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(STATISTICS)
STATISTICAL INFERENCE
COMPLEMENTARY COURSE
B.Sc. MATHEMATICS
III SEMESTER
(2011 Admission)
UNIVERSITY OF CALICUT
SCHOOL OF DISTANCE EDUCATION
CALICUT UNIVERSITY P.O., MALAPPURAM, KERALA, INDIA - 673 635
417
School of Distance Education
UNIVERSITY OF CALICUT
SCHOOL OF DISTANCE EDUCATION
STUDY MATERIAL
B.Sc. MATHEMATICS
(2011 Admission onwards)
III SEMESTER
COMPLEMENTARY COURSE
(STATISTICS)
STATISTICAL INFERENCE
Prepared by:
Dr.K. ANEESH KUMAR
Department of Statistics
Mahatma Gandhi College, Iritty
Keezhur .P.O. – Kannur – 670 703.
Scrutinised by:
Dr. ANIL KUMAR .V.
Reader & Head
Department of Mathematics
University of Calicut
Layout & Settings:
Computer Cell, SDE
©
Reserved
STATISTICAL INFERENCE
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CONTENTS
1.
SAMPLING DISTRIBUTIONS
Sampling distributions
Chi-square distribution
Student’s t-distribution
Snedecor’s F-distribution
5 – 29
2.
THEORY OF ESTIMATION-I (POINT ESTIMATION)
Statistical Inference
Point estimation
Desirable properties of good estimator
Methods of estimation
30 - 51
3.
INTERVAL ESTIMATION
52 – 68
Interval estimation
Confidence interval for the mean of a normal population
Confidence interval for the difference of means
Confidence interval for the variance of a normal population
Confidence interval for large samples
4.
TESTING OF HYPOTHESIS
Statistical hypothesis
Testing of hypothesis
Errors in testing of hypothesis
Steps in testing of hypothesis
Most powerful test
Neymaan-Pearson theorem
5.
LARGE SAMPLE TESTS
Large sample tests
Testing mean of a population
Testing the inequalities of means of two populations
Testing the population proportion
Testing the equality of proportion of two populations
Goodness of fit
2 test of independence
6.
SMALL SAMPLE TESTS
Small sample tests
Tests based on normally distributed test statistics
Tests based on statistics following t – distribution
Tests based on statistics following 2 - distribution
Tests based on statistics following F - distribution
STATISTICAL INFERENCE
69 – 89
90 – 124
125 – 139
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CHAPTER 1
SAMPLING DISTRIBUTIONS
1.1. Sampling Distribution
The characteristics of a large group of individuals are studying in any statistical
investigation. This group is referred to as the population under investigation. Let us
study only one characteristic, say X – the height of the individuals in the population.
Corresponding to each individual of the population we get a number denoting magnitude
of the characteristic considered.
The set of such numbers are called the statistical
population. This statistical population is considered as the set of admissible values of the
variable X (here height). The distribution of the values of X is known as the distribution of
this statistical population (or simply saying population now onwards).
Any function of the statistical population values are called population parameter.
For eg., mean, variance, median etc., of the variable considered.
The process of making inference about the population based on samples taken from
the population is known as statistical inference.
Consider a random sample taken from a population then the function of sample
values like sample mean, sample variance sample moments etc., are known as statistic.
The distribution of statistic is known as the sampling distribution of that statistic.
Sampling distribution of means of random samples taken from a normal population
N ( , ) :
Let X 1 , X 2 ,..., X n be the random samples taken from N ( , ) . Then,
Sample mean X
mgf of X , M
X 1 X 2 ... X n
n
t
(t ) M X1 X 2 ... X n (t ) M X1 X 2 ... X n ( )
X
n
n
t
t
t
M X1 ( ) M X 2 ( )....M X n ( )
n
n
n
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( X i ' s are ind .)
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We have for X~ N ( , ) ,
M X (t ) e
t
t 2 2
2
t t 2 2
t t 2 2
t t 2 2
2
2
2
(t ) e n 2n .e n 2n ....e n 2n
Therefore,
M
X
e
(
population with parameters and
t t 2 2 n
t 2 2
)
t
n 2n 2
2n , it is the m.g.f. of a normal
e
. This implies that
n
N ,
√
Problem 1: A random sample of size 25 is taken from a normal population with mean 1 and
variance 9. What is the probability that the sample mean is negative?
Solution:
Given sample size n = 25, 1 and 3 .
We
have
X ~ N ( ,
).
n
The required probability = P( X 0)
=P
=P
√
<
< − /
√
√
, where Z =
= P (Z < -1.67) = P(Z > 1.67)
N(0, 1)
√
= P (Z > 0) – P (0 < Z < 1.67)
( Standard normal curve is symmetric about the y-axis)
= 0.5 – 0.4525 (from standard normal table)
= 0.0475
========
Problem 2: A random sample is taken from a normal population with mean 10 and variance 9.
How large a sample should be taken if the sample mean is to lie between 9 and 11 with probability
0.95?
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Solution:
To find the minimum number of samples so as P(9 X 11) 0.95
Let n samples are taken from the given population N (10, 3),
then we have, X ~ N (10,
X 10
~ N (0,1)
3
n
ie., Z
P (9 X 11) 0.95
P(
ie., P (
9 10 X 10 11 10
) 0.95
3
3
3
n
n
n
1
1
nZ
n ) 0.95
3
3
ie., P (0 Z
For
Z
~
3
)
n
N
1
n ) 0.475
3
(0,1),
P (0 Z
1
n ) 0.475
3
happens
when
1
n 1.96 n 5.88 n 34.58
3
So the minimum number of sample for the required probability condition is 35.
1.2. Chi-square distribution ( 2 -distribution)
A random variable X with pdf,
1 2 x n 1
2 e 2 x2 , 0 x
f ( x) n2
0, otherwise
n
is said to follow 2 distribution with n degrees of freedom, where n is the parameter of
the distribution, denoted by X ~ 2 (n) .
Moment generating function of X ~ 2 (n) :
Observe that, M
STATISTICAL INFERENCE
X
(t ) E (etX )
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n
1
2 x n 1
etx 2 e 2 x 2 dx
n
0
2
n
1 2 x( 1 t ) n 1
2
2
x 2 dx
e
n 0
2
n
n
1 2
1 2
n
√
2
2
2
. (
=∫
n
n
n
1 t 2
2 1 t 2
2
2
dx )
This implies that
Mean and Variance:
Mx (t) = ( −
)
Observe that,
E( X )
=
=
d
M (t )
dt X / t 0
( −
( −
Hence, E(X) = n
Moreover, E(X2) =
= −
( −
)
− −
= n(n + 2).
)
t=0
)
−
t = 0 = n.
t=0
( −
)
−
t=0
Hence, V(X) = n(n+2) – n2 = 2n
Additive property of 2 distribution:
Let
X 1 and
X 1 ~ 2 ( n1 ) and X 2 ~ 2 ( n2 ) .
X 2 are two independent 2 random variables where
Then X 1 X 2 follow chi-square distribution with n1 n2
degrees of freedom.
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Proof: Observe that
X 1 ~ (n1 )
2
X 2 ~ ( n2 )
Therefore, M X X (t ) M X (t ) M X
1 2
1
2
1 2t
and
n
2
M X (t ) 1 2t 2 .
2
2
n
1
M X (t ) 1 2t 2
1
n
n
1
2
2
(t ) 1 2t
1 2t 2
n n
1 2
2
This is the m.g.f. of a 2 random variable with n1 n2 degrees of freedom. Hence
X 1 X 2 follow chi-square distribution with n1 n2 degrees of freedom.
In general if, X1, X2, ……Xk are n independent random variables with n1, n2, …… nk
degrees of freedom respectively, then X1+ X2 + ……Xk follow chi-square distribution with
n1+ n2 + ……nk degrees of freedom.
Tables of chi-square distribution:
Chi –square table gives the values of 2( ) for a 2 variable with various degrees
of freedom and for various values of , such that P ( 2 2( ) ) .
Theorem: If X ~ N (0, 1), then Y X 2 follow chi-square distribution with one degree of freedom.
Proof:
Observe that
M Y (t ) M X 2 (t ) E (etX )
2
e
STATISTICAL INFERENCE
2
tx2
1 x2
e dx
2
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1
2
e
tx2
x2
2 dx
( e
put x 2 u dx
This implies that
1
2
e
tx2
x2
2 dx
0
x2
2
is an even function)
1
x2 (t )
2 dx
0
2
2
1
2
e
tx2
.
du
du
2x 2 u
M Y (t )
But (1 2t )
2
2
2
2
e
1
u (t )
2
0
du
2 u
u ( 1 t ) 1 1
e 2 u 2 du
0
My(t) = ( −
)−
1
2
1
.
1
2 1
2
( t)
2
.
is the m.g.f. of a chi-square random variable with one degree of freedom.
Hence Y 2 1 .
Theorem: If X 1 , X 2 ,..., X n are n random samples taken from a standard normal population, then
the sum of squares of random sample follows 2 n
Proof:
Since X i ' s are random samples from N (0, 1), they are independent and
X i 2 ~ 2 (1) for all i . If X follows N (0, 1), we have X 2 ~ 2 (1) .
Here X i 2 ~ 2 (1) for all i .
Hence, by additive property of chi-square distribution, the
sum of squares of X 1 , X 2 ,..., X n follow 2 n .
Theorem:
If
X 1 , X 2 ,..., X n are n random samples taken from N ( , ) ,
Show that
x
2
Y i
~ ( n)
i 1
n
2
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Proof:
Given X 1 , X 2 ,..., X n are independent samples from N ( , ) .
Then
,
,...
,
N (0, 1)
,...,
each follow 2 (1)
Since X 1 , X 2 ,..., X n are independent, by additive property of chi-square distribution,
xi
2
~ ( n) .
i 1
2
n
we get,
Problem 1: If X 1 , X 2 ,..., X n are n random samples taken from N ( , ) , find the distribution of
sample variance S 2 .
Solution:
For
the
random
samples
X 1 , X 2 ,..., X n
taken
from N ( , ) ,
we
have
x
2
Y i
~ (n) . Note that Y can be written as
i 1
n
2
2
x X X
Y i
, where X is the sample mean
i 1
n
(x X ) ( X )
ie., Y i
i 1
n
2
( xi X ) 2 ( X ) 2 2( xi X )( X )
2
i 1
n
n
i 1
n
i 1
( xi X ) 2 n ( X ) 2 n 2( xi X )( X )
2
2
2
i 1
i 1
( xi X ) 2 n( X ) 2
2
2
2
n
(x X )
X
i 2
i 1
n
2
n
( ( xi X ) 0)
i 1
2
nS
X
2
n
where
STATISTICAL INFERENCE
2
X
~ N (0,1)
n
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Observe that
X
n
2
N(0, 1) and
√
follow chi-square distribution with one degree of freedom. Since X and S 2 are
independently distributed, then by additive property of chi-square distribution
nS 2
2
~ 2 (n 1) .
Let U =
nS 2
~ 2 (n 1)
2
(1)
Then,
1 n21 u n 1
1
2
2 2
,
f (u ) n 1 e u
2
0, otherwise
Equation (1) can be written as S2 =
S2
u 2
, then,
n
, therefore f (S2) = f(u)
f( S 2 )= f(u) in terms of S 2
n1
1 2
f (S 2 ) 2
n1
1 2
2
e
n 1
2
n1
Hence,
f (S
n 2
) 2
2
n 1
2
nS 2
2 2
nS 2
2 2
e
n 1
2
2
0u
e
nS 2
2 2
du
dS 2
nS 2
2
nS 2
2
S2
n 1
1
2
n 1
1
2
n 1
1
2
,
du
dS 2
n
.
2
0 S2
Problem 2: If X ~ 2 (n) , find the mode of X.
Solution:
n
We have
STATISTICAL INFERENCE
1 2 x n 1
f ( x) 2 e 2 x 2
n
2
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Mode is the point x where f(x) attains its maximum. That is, the point x, where
f ' ( x) 0 and f '' ( x ) 0 .
n
'
f
x
n
x n
1 2
2 2 2 2 1
2 2 n
( x)
e
1 x
e x
n
2
2
x n n 2 x n 1
f ' ( x) 0 e 2 1 x 2 e 2 x 2 0
2
e
x
2
n
n
n
1
n 2 2
x2
1 x
2
x
n
1
n 2 2
e 2 x2
1 x
2
x 1
1
n2
x n2
For the value of x = n-2, f '' ( x ) 0 . Hence the mode is at x = n-2.
Problem 3:
If
X
is
distributed
as
f ( x)
1
, 0 x ,
show
x
that 2 log e ( )
follows chi-square distribution with 2 degrees of freedom.
Solution:
Let
x
Y 2 log e ( ) . This can be written as x =
Note that
−.
=
f(y) = f(x) in terms of y .
dx
dy
= . .
1 2y
e
2
2 1
=
12 2
21
2
e
y
2
2
1
2
y
f ( y)
1
2
2 1
2
21
2
e
y
2
2
1
2
y ,
0 y ;
This implies that Y X(2)
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Problem 4: For large n, show that chi-square distribution approximately normally distributed.
Solution:
Consider a random variable X following chi-square distribution with n degrees of
freedom.
Then
Consider Z
M X (t )
E( X ) n
1 2t
and
. More over,
V(X) =2n.
X n
, then
2n
M (t ) M
(t ) e
Z
X n
2n
e
Therefore,
n
2
n
t
2n M
X
(
t
)
2n
n
t
n
2n 1 2 t 2 .
2n
n
n
2n t
t 2
log M (t ) log e
1 2
Z
2n
n
t
t n log 1 2
2
2n
2n
2
n
t
1
t
n
t 2
2
.....
2
2n
2n 2
2n
1
n
n
t2
t
t ... + (many terms involving n 2 and its
2
2n
2n
higher power in denominator)
As n become very large,
That is
t2
log M (t )
.
Z
2
t2
2
M (t ) e . This is the m.g.f. of a standard normal random
Z
variable. Then by uniqueness theorem of m.g.f., X~N(n, 2n ) for large n.
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Problem 5: For a random sample of size 16 from N ( , ) population, the sample variance is 16.
Find a and b such that P( a 2 b ) 0.60
Solution:
P( a 2 b ) 0.60 implies P (
P(
1
1
1
2 ) 0.60
a
b
nS 2 nS 2 nS 2
nS 2
2
) 0.60, where 2 ~ 2( n1) .
a
b
Putting S 2 =16 and n = 16 in the above equation, we get
P(
That is,
16 16
16 16
2(161)
) 0.60
a
b
P(
16 16
16 16
2(15)
) 0.60
b
a
--------- (1)
From the table of chi-square distribution,
P ( 2 (15) 10.307 ) 0.80 and P ( 2(15) 19.311) 0.20 .
Therefore,
P (10.307 2 (15) 19.311) 0.60
Comparing (1) and (2), we get
Hence
b
----------(2)
16 16
16 16
10.307 and
19.311
b
a
16 16
16 16
24.84 , and a
13.26
10.307
19.311
ie., P( 13.26 2 24.84 ) 0.60 .
1.3 Student’s t-distribution
This is the probability distribution which was introduced by W.S Gossset and
known in his pen name ‘student’. A continuous random variable t with density function
n 1
n 1
2 2
t
2
f (t )
, t
1
n
n
n
2
is said to follow student’s t-distribution with n
degrees of freedom.
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Note that if X ~ N (0,1) and Y ~ 2 (n) , then t
X
Y
n
follows t-distribution with n degrees
of freedom, where X and Y are independent.
Examples of statistics following student’s t-distribution:
Let X be the mean of n random samples taken from N ( , ) . Let S 2 be the
1.
sample variance. Then we have X ~ N ( ,
X
~ N (0,1) .
n
Note that
Therefore,
t
This implies that
2.
nS 2
) and 2 ~ 2 (n 1) .
n
X
n
~ t( n 1) .
nS 2
2
(n 1)
t
( X ) n 1
~ t( n 1) .
S
Let X 1 and X 2 be the means, S1 and S 2 be the standard deviations of
samples of sizes n1 and n2 taken independently from two normal populations with same
mean and standard deviation . Then,
t
Proof:
X1 X 2
n1S 1 n2 S 22 1 1
n1 n2 2 n1 n1
2
We have
This implies that
STATISTICAL INFERENCE
X1 X 2
1 1
n1 n2
2 2
n1
1
2 2)
) and X 2 ~ N ( ,
)
n1
n2
X 1 ~ N ( ,
Then, X 1 X 2 ~ N (0,
~ t( n n
n2
).
~ N (0,1)
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n1S12
n2 S 2 2
2
~
(
n
1)
and
~ 2 (n2 1) .
1
2
2
Also we have,
Using additive (reproductive) property of chi-square distribution, we get
n1S12 n2 S 2 2
2 ~ 2 (n1 n2 2)
2
So,
t
That is,
t
That is,
t
n1S12
n2 S 2 2
( 2 and
are ind .)
2
X1 X 2
1 1
n1 n2
~ t( n n
n1S12 n2 S 2 2
2
2
n1 n2 2
1
n1 n2 2
X1 X 2
1 1
n1S 12 n2 S 22
n1 n2
X1 X 2
n1S 12 n2 S 22 1 1
n1 n2 2 n1 n1
2 2)
~ t( n n
1
2 2)
~ t( n n
1
2 2)
Tables of t-distribution:
Note that t-distribution is symmetric about zero and bell shapped. Tables of tdistribution gives the values of t for various degrees of freedom and for various value of
, such that P(| t | t ) .
Problem 1: If t ~ t( n ) , find the mode of t.
Solution:
We have,
STATISTICAL INFERENCE
n 1
n 1
2 2
2 1 t
f (t )
, t
n
n
n
2
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Mode is the point t where f(t) attains its maximum.
That is the point t, where
f ' (t ) 0 and f '' (t ) 0 .
f ' (t )
n 1
n 1
2 2 1
n
1
t
2t
2
.
1
n
n
n 2
n
2
n 3
2 2
t
0
f (t ) 0 2t 1
n
'
t0
f '' (t )
n 1
n 3
n 3
2 2 1
2 2
n
1
2
n
3
t
t
.
2
t 1
1
n
n
n 2 n 2
n
2
n 1 n 1
2 n
f '' ( x)
0 . Hence Mode of is at t = 0.
n
n
2
At at t 0,
Problem 2: If t ~ t( n ) , then as n , prove that t ~ N (0,1) .
Solution:
Given t ~ t( n ) , we have
n 1
n 1
2 2
t
2
f (t )
, t
1
n
n
n
2
Using the following results we can prove as n , t ~ N (0,1)
(i)
as n becomes very large,
(ii)
lim 1
n
n
n
nk
nk
n
e
Note that as n
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1
n 2
1
2 t2 2
f(t)
e
n
which is the probability density function of standard normal distribution.
Problem 3: If t ~ t(5) , find a such that, P ( a t a ) 0.98
Solution:
The graph of t ~ t(5) is symmetric about zero. To find a such that the area under
the t curve between –a and a is 0.98.
ie., to find a such that P( t a) 0.98 or P( t a) 0.02
From table of t-distribution, for 5 d.f. we get, a = 3.365.
Problem 4:
Prove that the ratio of two independent standard normal random variables is a
student’s t random variable with 1 degree of freedom.
Solution:
Given X 1 ~ N (0,1) , X 2 ~ N (0,1) and they are independent.
Then, X 22 ~ 2 (1) . This implies that
t
X1
X2
~ t(1)
2
1
That is,
t
X1
~ t(1)
X2
That is, the ratio of two independent standard normal variables follows t(1) .
Problem 5: If X 1 and X 2 are two independent standard normal variables, find the distribution
of t
2X 1
X X 22
2
1
.
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Solution:
Given X 1 ~ N (0,1) , X 2 ~ N (0,1) and they are independent.
So, X 12 ~ 2 (1) and X 22 ~ 2 (1) .
By additive(reproductive) property of chi-square distribution,
X 12 X 22 ~ 2(2) .
Then,
That is,
t
t
X1
X X 22
2
2
1
2X 1
X 12 X 22
~ t( 2 ) .
follow t-distribution with 2 degrees of freedom.
Problem 6: Find the maximum difference that we can expect with probability 0.95 between the
means of samples of sizes 10 and 12 from a normal population, if their standard deviations are
found to be 2 and 3 respectively.
Solution:
Let x1 and x2 be the means of the samples of sizes n1 =10 and n2 =12 taken
randomly from two normal populations. Assume samples are taken independently. The
sample variances S12 = 4 and S 2 2 = 9 respectively.
To find the value of k such that,
P x1 x2 k 0.95 .
For samples taken from two normal populations independently, we
know that,
X1 X 2
n1S 12 n2 S 22 1 1
n1 n2 2 n1 n1
~ t( n n
2 2)
1
Then to find k such that,
P
X1 X 2
n1S1 n2 S 22 1 1
n1 n2 2 n1 n1
STATISTICAL INFERENCE
2
k
0.95
n1S 12 n2 S 22 1 1
n1 n2 2 n1 n1
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That is, P t(1012 2)
That is,
k
0.95 .
10 4 12 9 1 1
10 12 2 10 12
k
P t( 20 )
0.95
1.165
The table of t- distribution for 20 d.f. ,
Comparing (1) and (2), we get,
(1)
P t( 20) 2.0860 0.95 ---- (2)
k
2.086 . Hence
1.165
k 2.086 1.165 2.431
That is the maximum difference that can expect with 95% probability is 2.431.
1.4. Snedecor’s F-distribution
A continuous random variable F with pdf
n1
2
n
1 1
n1
2
F
n2
f (F )
(
n1 n2
2
, 0F
n1 n2 n1
, ) 1 F
2 2 n2
is said to follow F-distribution with (n1 , n2 ) degrees of freedom.
If X 1 and X 2 are independent random variables following chi-square distribution
with n1 and n2 degrees of freedom respectively, then,
X1
F
X2
n1
~ F (n1 , n2 )
n2
Statistic following Snedecor’s F-distribution:
1.
Let independent samples of sizes n1 and n2 are taken from normal population with
mean and standard deviation . Let S12 and S 22 are the respective sample variance,
then F
n1S12 (n2 1)
~ F (n1 1, n2 1)
n2 S 22 (n1 1)
STATISTICAL INFERENCE
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Proof:
For the set of samples taken from normal population, we have,
n1S12
~ 2( n 1) and
1
2
n2 S22
~ 2( n 1)
2
2
n1S12
2
Then,
F
n1 1
2
2 2
2
nS
~
F (n1 1, n2 1)
n2 1
n1S12 (n2 1)
Hence F
~ F (n1 1, n2 1)
n2 S 22 (n1 1)
Tables of F-distribution:
Tables of F-distribution gives the values of F for various values of n1 , n2 and
,
such that P( Fn ,n F ) .
1 2
Mode of F-distribution:
Mode is the point F where f(F) attains its maximum. That is the point F, where
2 log f ( F )
log f ( F )
0.
f ' ( F ) 0 and f '' ( F ) 0 or the F where
0 and
F 2
F
n1
We have,
f (F )
n
n1 2 21 1
F
n2
n n n
( 1 , 2 ) 1 1 F
2 2 n2
Therefore,
log f ( F )
STATISTICAL INFERENCE
n1 n2
2
n n
n
n1
n n
n n
log 1 1 1 log F log ( 1 , 2 ) 1 2 log 1 1 F
2
2 2
2
n2 2
n2
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School of Distance Education
i.e.,
n
log f ( F ) n1
1
1 n n
1 1 2 .
. 1
F
2
n1 n2
2
F
1 F
n2
1
n
1 n n n
1 1 1 2 1 .
2
n2 n1F
2
F
log f ( F )
0
F
F
n
1
1 1
2
F
n1 n2 n1 .
n2 n1 2
.
n1 n2 2
at this point it can be verified that
Hence mode of F ~ F (n1 , n2 ) is F =
Remark:
The mode
2
1
n2 n1F
2 log f ( F )
0.
F 2
n2 n1 2
.
n1 n2 2
n2 n1 2
can be expressed as
n1 n2 2
n 2 . Since F 0 , the
n2
1
n1
n2 2
mode cannot be negative. Hence n1 should not be less than 2. So the mode exists only
when n1 2 . Again since
n2
n 2 are les than 1, the mode is always less than
and 1
n1
n2 2
unity.
Problem 1:
Prove that the ratio of the squares of two independent standard normal random
variables is an F- random variable with (1, 1) degree of freedom.
Solution:
Let X 1 ~ N (0,1) , X 2 ~ N (0,1) and they are independent
So, X 12 ~ 2 (1) and X 22 ~ 2 (1)
X 12 1
~ F (1,1)
X 22 1
then
ie.,
STATISTICAL INFERENCE
X 12
~ F (1,1) .
X 22
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Problem 2: If X is a random variable following F distribution with n1 , n2 degrees of freedom.
Find the distribution of Y
1
.
X
Solution:
Given Y
1
we have f(y) = f(x) in terms of y .
X
n1
2
Here
(
Y
1
X
, so
X
n1
1
x2
n1
n2
f ( x)
n1 n2
2
, 0 x
n1 n2 n1
, ) 1 x
2 2 n2
1
dx
1
2
Y
dy
y
n1
2
n1
1
2
n1
1
n2 y
so, f ( y )
(
n1 n2
2
n1 n2 n1 1
, ) 1
2 2 n2 y
n1
n n n y n1
( 1 , 2 ) 2
2 2 n2 y
n1
2
(
n1 n2
2
n1
2
n1 n2
2
1
y
n1
n n
1 1 2
2
2 2
n1
1
n2 y
(
2
n n
( 1 2)
2
n1 n2
n
, ) y 1
2 2
n2
1
y2
1
y
n1
1
2
n1
1
n2 y
n1
1
n1 2 1 2
n2 y
f ( y)
STATISTICAL INFERENCE
dx
dy
n1 n2
2
n1 n2 yn2
, )
1
2 2 n1
n1 n2
2
n2
n1
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n1 n1 n2
2 2
n
1 2
2
n1 2
n2
1
y
n n n
( 1 , 2 ) 1 2
2 2 n1
n2
2
n2
n1
f ( y)
(
y
n2
1
y2
n1 n2
2
n1 n2
n2 n1
( 2 , 2 ) ( 2 , 2 )
n1 n2
2
n2 n1 n2
, ) 1 y
2 2 n1
Y ~ F (n2 , n1 )
Problem 3: If X following F distribution with n1 , n2 degrees of freedom Y follow F distribution
with n2 , n1 degrees of freedom. Prove that P ( X c ) P ( Y
1
).
c
Solution:
P ( X c ) P(
1
1
)
X
c
But, given, X ~ F (n1 , n2 ) , then,
1
~ F ( n2 , n1 )
X
Also Y is a variable following F (n2 , n1 ) .
Hence,
P ( X c ) P (Y
1
)
c
Problem 4: If X following F distribution with n , n degrees of freedom. If , ( ) are
such that P ( X ) P ( X ) . Show that . 1
Solution:
Since X ~ F ( n, n) ,
1
also ~ F (n, n) ----(1)
X
P ( X ) P(
1
1
)
X
Given P ( X ) P ( X )
P(
STATISTICAL INFERENCE
1
1
) P( X )
X
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School of Distance Education
P( X
1
1
) P( X )
(by (1))
1
Problem 5: If t follows student’s t-distribution with n degrees of freedom, prove that t 2 follows F
distribution with 1 , n degrees of freedom.
Solution:
Given t ~ t( n ) , we have
Let Y= t 2 ; t
n 1
n 1
2 2
t
2
f (t )
, t
1
n
n
n
2
Therefore
,
Y
dt
dy
f(Y) = f(t) in terms of y .
Note that
1
2
Y
dt
dy
n 1
n 1
Y
2 1
f (Y ) 2. 2 1
.
, 0Y
n n
2 Y
n
2
That is,
1
2. .
2
f (Y )
n 1
n 1
1
Y
2
2 .Y 2
1
1 n n
n
2 2
n 1
n 1
2 1
Y
2
.Y 2
1
1 n n
n
2 2
1
Y2
1
1 n Y
n , 1
2 2 n
STATISTICAL INFERENCE
n 1
2
mn
m , n
m n
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1
1
1 2 2 1
.Y
n
f (Y )
1 n 1
, 1 .Y
2 2 n
n 1
2
, 0 Y ,
this is the density
function of a random variable following F-distribution with (1, n) degrees of freedom.
Hence Y t 2 follows F (1, n) .
Problem 7: If X following F distribution with n1 , n2 degrees of freedom, prove that as n2
, Y= n1 X follows chi-square distribution with n1 degrees of freedom.
Solution:
Given Y= n1 X,
dx
1
;
dy
n1
f(Y) = f(x) in terms of y .
n1
2
X ~ F( n ,n
1 2)
;
n1
n2
f ( x)
(
dx
dy
n1
1
x2
n1 n2
2
, 0 x
n1 n2 n1
, ) 1 x
2 2 n2
n1
2
n1
n2
f ( y)
(
n1
1
x2
n1 n2
2
1
n1
1
n1
n1 n2 n1 y
, ) 1
2 2 n2 n1
n1
1
n1
y 2
n1 n2
n
1 2
2
n1
n1
n2 2
STATISTICAL INFERENCE
n1 n2
2
n1 n2
y
1
2 2 n2
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School of Distance Education
As n2
n1
n2 2
2
n
1
n2 2
n1 n2
2
n1
n
n2 2 2
2
n1 n2
2
y
1
n2
Note that
Also note that
lim
as n2 ,
n2
1
n1
22
nk
nk
n
1
as
n
n1
n 2
y 2
y 2
1 1
n2 n2
n1 n2
2
1
n1
n 2
y 2
y 2
lim 1 lim 1
n2
n2 n2 n2
y
1
n2
y
2
e
n
y2
lim 1 e y
n2 n2
1
n1
1
Hence, as n2 ,
y 2
n
f ( y) 1
n1
n1
2
e
y
2
1
n1
n1
2 2 n1
2
n1
1
1 2
n
1
n1
2
n1 y
n1
22
Hence,
f ( y)
1
n1
2 2 n1
e
y
n1
1 2
2
e
1
n1
n1
2
y n1
1
2 .y 2
,
0 y
2
That is Y follow chi-square distribution with n1 degrees of freedom.
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EXERCISES
1. Explain what is meant by sampling distribution. State the relationship between
normal and chi-square distribution.
2. Define chi-square distribution with n degrees of freedom. Derive its mean and
variance.
3. State and prove the reproductive property of chi-square distribution.
4. Show that for Students t-distribution with n degrees of freedom, the mean
n 1
n 2
deviation is given by
.
n
2
5. Define F-distribution. Explain its use in statistical inference.
6. State the inter-relationship of t, chi-square and F distributions. A random variable
X has F- distribution with (n, m) degrees of freedom. Find the distribution of Y
1
X
.
7. Derive Student’s t- distribution and establish its relation with F- distribution.
8. If F has F-distribution with (n, m) degrees of freedom, prove that as n , nF tends
to be distributed as chi-square with n degrees of freedom.
9. If X and Y are independent standard normal variables, find the distribution of
Z
X2
and write down its p.d.f.
Y2
10. X 1 , X 2 , and X 3 are independent N(0,1) variables. Find the distribution of
(i ) X 1 2 X 2 2
(ii )
X2
X1
and (iii )
X 12 X 2 2
2 X 32
****
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CHAPTER 2
THEORY OF ESTIMATION- POINT ESTIMATION
2.1. Statistical Inference
Making inferences about the unknown aspects of the population using the samples
drawn from the population is known as statistical inference. The unknown aspects may
be the form of the probability distribution of the population or values of the parameters
(ie., function of population values) involved, or both.
Two important subdivision of statistical inference are.
(i)
Estimation
(ii)
Testing of hypothesis.
Estimation of parameters:
The theory of estimation was founded by Prof.R.A.Fisher, who is known as the
father of modern Statistics. Estimation deals with function of sample values, the value of
which may be taken as the values of the unknown parameters (known as point estimation)
as well as with the determination of the intervals which will contain the unknown
parameters with a specified probability (known as interval estimation), based on the samples
taken from the population.
Testing of hypothesis deals with the method of deciding whether to accept or reject
the hypothesis regarding the unknown aspects of the population, based on the samples
taken from the population.
2.2. Point estimation
In point estimation a number is suggested as a value of the unknown parameter,
using the values of the sample observations taken randomly from the population. The
function of sample values, suggested as a good approximation for the required parameter
is known as an estimator, and a particular value of the estimator is known as the estimate.
For eg., to estimate population mean, sample mean is taken as an estimator and the value
of sample mean of a particular sample is an estimate of population mean.
2.3. Desirable Properties of Good Estimator
An estimator of a parameter is said to be a good estimator if it satisfied some desirable
properties. They are
(i) Unbiasedness
(ii) Consistency (ii) Efficiency (iv) Sufficiency
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(i) Unbiased ness:
Let x1 , x2 ,..., xn are random samples taken from a population with unknown parameter .
The statistic tn t ( x1 , x2 ,..., xn ) is said to be an unbiased estimator of , if E (tn ) .
tn is an
unbiased estimator of a function of , say f ( ) , if E (tn ) f ( ) .
Problem 1: A random sample x1 , x2 ,..., xn is taken from a population with mean . Show that the
sample mean x is an unbiased estimator of .
Solution:
Since the samples are taken from a population with mean ,
E ( x1 ) E ( x2 ) ... E ( xn )
we have x
x1 x2 ... xn
n
x x ... xn 1
E(x ) E 1 2
E ( x1 x2 ... xn )
n
n
1
1
( ... ) = (n )
n
n
E(x)
Hence x is an unbiased estimator of
Remark: Unbiased estimator for a parameter need not be unique. For eg. in the above
case
consider
the
first
two
observations x1 and x2
only.
Then
x x
1
x x 1
E 1 2 E ( x1 x2 ) E ( ) . That means 1 2 is also an unbiased
2
2
2 2
estimator of . In similar way we can find many unbiased estimators for .
Problem 2: A random sample x1 , x2 ,..., xn is taken from a normal population with mean and
standard deviation 1. Show that t
1 n 2
xi is an unbiased estimator of 2 1 .
n i 1
Solution:
1 n
E (t ) E xi 2
n i 1
1
E ( x12 ) E ( x2 2 ) ... E ( xn 2 )
n
Given the population variance as 1, and population mean as ,
E ( xi ) and
STATISTICAL INFERENCE
E ( xi 2 ) E ( xi ) 1 for all i
2
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1 E ( xi ) E ( xi 2 ) for all i
2
Hence,
E ( xi 2 ) 1
2
E (t )
E (t )
for all i
1
1 2 1 2 ... 1 2
n
1
n 1 2
n
1 ,
ie., t
2
1 n 2
xi is an unbiased estimator of 2 1 .
n i 1
Problem 3: For the random sample x1 , x2 ,..., xn taken from N ( , ) , show that the sample
variance is a biased estimator of the population variance.
Solution:
Here to show that E ( S 2 ) 2 ,
samples x1 , x2 ,..., xn taken from N ( , ) .
Note that
where S 2 is the sample variance of the random
E ( S ) S 2 f ( S 2 )dS 2
2
0
n 1
i.e,
n 2
nS 2
n 1
2
1
2
2
2
2 2
2
f (S )
e
(S )
, 0 S2
n 1
2
n 1
n 2
nS 2
n 1
2
1
2
2
E (S 2 ) S 2
e 2 ( S 2 ) 2 dS 2
n 1
0
2
n 1
n 2
2
2
n 1
2
e
nS 2
2 2
(S
2
n 1
)2
dS 2
0
n 1
n 2
2
2
n 1
2
STATISTICAL INFERENCE
e
nS 2
2 2
(S 2 )
n 1
1
2
dS 2
0
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n 1
n 2
n 1
2
2
2
n 1
n 1
n
2
2
2
2
e
mx
( x) n 1 dx
0
n
n
m
n 1
n 2 n 1 n 1
1
1
2
2
2
2
n 1
1
n 1
n 2
2
2
2
n 1 2
2 n
2
n 1
2
E (S 2 )
n
Hence S 2 is not an unbiased estimator of 2 .
That is S 2 is a biased estimator of 2 .
n 1
2
E (S 2 )
n
Here,
That is,
E(
nS 2
) 2
n 1
nS 2
is an unbiased estimator of 2 .
n 1
Problem 4: For the random sample x1 , x2 ,..., xn taken from Poisson population with parameter ,
obtain an unbiased estimate of e .
Solution:
Consider a statistic t defined as follows,
t 1, if the first observation of the sample if zero
= 0, otherwise.
P(the first observation of the sample if zero) =
e 0
e
0!
Hence, E (t ) 1 e 0(1 e ) e
This implies the statistic t is an unbiased estimator of e .
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Problem 5: For the random sample x1 , x2 ,..., xn taken from B (1, p ) , show that
T (T 1)
is an
n(n 1)
n
unbiased estimator p 2 , where T xi .
i 1
Solution:
Here x1 , x2 ,..., xn are from B (1, p ) . Then by the additive property,
T x1 x2 ... xn follows B ( n, p )
Note that
E (T ) np , V (T ) npq
T (T 1)
1
1
E
E T (T 1)
E T 2 T
n(n 1)
n(n 1) n(n 1)
E (T 2 ) V (T ) E (T )
2
npq n 2 p 2
T (T 1)
1
E
E npq n 2 p 2 np
n(n 1)
n(n 1)
1
E np (1 p ) n 2 p 2 np
n(n 1)
=
1
E n 2 p 2 np 2
n(n 1)
T (T 1)
T (T 1)
E
p 2 , ie.,
is an unbiased estimator of p 2 .
n(n 1)
n(n 1)
( ii ) Consistency:
The
Let x1 , x2 ,..., xn are random samples taken from a population with unknown parameter .
statistic tn t ( x1 , x2 ,..., xn ) is said to be a consistent estimator of , if
P tn 1 as n or
tn is a consistent estimator of a function of , say f ( ) , if
P tn f ( ) 1 as n .
In other words tn is a consistent estimator of , if tn converges to in probability, and is
p
denoted as tn
.
Sufficient conditions for consistency:
Let {tn } sequence of estimators of , and if,
(i) E (tn ) or , as n
STATISTICAL INFERENCE
and (ii) V (tn ) 0, as n
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Then
tn is a consistent estimator of .
Proof:
Consider the statistic tn , then by Tchebycheff’s inequality,
P tn t.SD (tn ) 1
Let t.SD(tn ) c , then t
1
t2
c
SD (tn )
P tn c 1
as n ,
if E (tn ) or ,
c
number and,
,
SD (tn )
1
c
SD (tn )
2
and V (tn ) 0 ; then t.SD(tn ) c
P tn c 1
becomes a small
p
ie., tn
Hence under the given conditions, tn is a consistent estimator of .
Problem 1: For the random sample x1 , x2 ,..., xn taken from N ( , ) , show that sample mean x
is consistent estimator of population mean .
Solution:
x
x1 x2 ... xn
n
as n , E ( x )
E ( x ) and V ( x )
2
n
and V ( x ) 0 .
Hence, x is consistent estimator of population mean .
Problem 2: For the random sample x1 , x2 ,..., xn taken from Poisson population with parameter ,
nx
show that
is consistent estimator .
n 1
Solution:
Here x1 , x2 ,..., xn
are taken from Poisson population with parameter , so
E ( X i ) and V ( X i ) for all i , then
x
x1 x2 ... xn
n
E ( x ) and V ( x )
n
Hence, as n ,
STATISTICAL INFERENCE
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n
nx
E
E(x )
n 1
n 1
1
E ( x ) E ( x ) , and
1
1
n
nx
V
n 1
n2
n 1
2
V (x )
1 n
1
n
1
2
0
nx
satisfies the sufficient conditions to be satisfied by consistent estimator
n 1
and hence it is a consistent estimator of .
Here
Problem 3: For the random sample x1 , x2 ,..., xn
consistent estimator of p (1 p ) , where T
taken from B (1, p ) , show that T (1 T )
is a
n
1
xi .
n i 1
Solution:
Note that x1 , x2 ,..., xn are from B (1, p ) . Then by the additive property,
X x1 x2 ... xn follows B ( n, p )
T
1 n
xi
n i 1
E (T ) E (
X
1
pq
) p , V (T ) 2 V ( X )
n
n
n
E (T 2 ) V (T ) E (T )
2
E (T 2 )
pq
p2
n
--------- (1)
pq
E T (1 T ) E (T ) E (T 2 ) p
p2
n
as n ,
np p (1 p ) np 2 np (1 p ) p (1 p )
n
n
n 1
p (1 p )
n
E T (1 T ) p(1 p)
V T (1 T ) E T (1 T ) E T (1 T )
STATISTICAL INFERENCE
2
--------- (2)
2
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pq
E T 2 T 4 2T 3 p
p2
n
2
--------- (3)
X 4
1
E T 4 E 4 E ( X 4 )
n n
1
n(n 1)(n 2)(n 3) p 4 6n(n 1)(n 2) p 3 7 n(n 1) p 2 np
n4
as n ,
E T 4 p 4
1
similarly , E T 3 4 n(n 1)(n 2) p 3 3n(n 1) p 2 np
n
as n ,
E T 3 p 3 and E T 2 p 2 (by (1))
Then, as n , from (3), we get,
V T (1 T )
p 2 p 4 2 p 3 p p 2
2
ie., V T (1 T ) 0; as n
Hence
--------- (4)
by (2) and (4) , T (1 T ) is a consistent estimator of p (1 p )
Problem 4: For the random sample x1 , x2 ,..., xn taken from N ( , ) , show that the sample
variance is a consistent estimator of the population variance.
Solution:
Let S 2 is the sample variance of the random samples x1 , x2 ,..., xn taken from N ( , ) .
Then we have.
n 1
n 2
nS 2
n 1
2
1
2
2
2
2 2
2
f (S )
e
(S )
, 0 S2
n 1
2
It is already found in a problem of last section,
n 1
2
E (S 2 )
n
Then, as n ,
E (S 2 ) 2
------- (1)
n 1
n 2
nS 2
n 1
2
1
2
2 2
2 2 2
2
E S S
e ( S 2 ) 2 dS 2
n 1
0
2
STATISTICAL INFERENCE
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n 1
n 2
2
2
n 1
2
e
nS 2
2 2
(S 2 )
n 1
2
dS 2
0
n 1
n 2
2
2
n 1
2
e
nS 2
2 2
(S 2 )
n 3
1
2
dS 2
0
n 1
n 2
n3
2
2
2
n 3
n 1
n
2
2
2
2
n 1
n 2 n 1 n 1 n 1
2
2
2 2 2
n 3
n 1
n 2
2
2
2
n2 1
4
n2
2
Hence, V ( S ) E S 2 E S 2
2
as n ,
2
n 2 1 4 n 1
2
2
n
n
V (S 2 ) 4 4 0
2
-------- (2)
From (1) and (2), it can infer that S 2 is a consistent estimator of the population
variance 2
Problem 5: Let t be a consistent estimator of , and let ( ) be a continuous function of .
Then prove that (t ) is a consistent estimator of ( ) .
Solution:
Since t is consistent for , P ( | t | ) 1 as n becomes large.
If is a continuous function, we have for such that, | t | ,
| (t ) ( ) | 1
P( | (t ) ( ) | 1 ) 1
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( iii ) Efficiency:
As we already seen, to estimate a particular population parameter there may exist
more than one unbiased estimators. Based on the variance of these unbiased estimators
their efficiency is defined. Consider t1 and t2 are two unbiased estimators of the
parameter . The estimator t1 is said to be more efficient than t2 , if Var( t1 ) < Var( t2 ). Let
t be the most efficient estimator for the parameter , then efficiency of any other unbiased
var(t )
estimator t1 of is defined as E (t1 )
. The efficiency of
var(t1 )
most efficient estimator is 1 and any other unbiased estimator is less than 1
The relative efficiency of t1 with respect to t2 is denoted by E and is defined as
var(t2 )
.
E
var(t1 )
Problem 1: For the random sample x1 , x2 ,..., xn taken from N ( , ) , test whether the following
statistics
are
unbiased
estimators
of
Which
one
is
more
efficient?
.
x x x x
x x
x 2 x2
(i ) 1 2 (ii ) 1
(iii ) 1 2 3 4
2
3
4
Solution:
x x
2
x x 1
(i) E 1 2 E x1 x2
,so 1 2 is unbiased estimator of .
2
2
2 2
x1 2 x2
1
x 2 x2 1
(ii) E 1
is an unbiased estimator of .
E x1 2 x2 2 , so
3
3 3
3
4
x x x x 1
(iii) E 1 2 3 4 E x1 x2 x3 x4
,
4
4
4
unbiased estimator of .
2 2
2
x1 x2 1
V
V x1 x2
4
2
2 4
so
x1 x2 x3 x4
4
also is an
( xi ' s are random samples)
1 2
5 2
x 2 x2 1
2
V 1
V
x
2
x
4
1
2
9
9
3 9
4 2 2
x x x x 1
V 1 2 3 4 V x1 x2 x3 x4
4
16
4
16
x x
x x x x
x 2 x2
Among these V 1 2 3 4 V 1 2 V 1
4
3
2
Hence
x1 x2 x3 x4
is more efficient.
4
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(iv ) Sufficiency :
In many problems of statistical inference, a function of the sample observations
contains as much information about the unknown parameter as do all observed values.
To estimate probability of head (p) when a coin is tossed, let the
coin is tossed n times and let xi 1 , if the i th toss is a success and =0¸otherwise. Then,
t xi - the total number of heads out of n tosses is enough to estimate p. It seems
i
unnecessary to know which toss resulted in a head.
That is t xi is sufficient to
i
estimate the parameter p. The result of n tosses x1 , x2 ,..., xn contains no other information
about p than that contains in t. Hence the conditional probability of x1 , x2 ,..., xn given
x
i
is independent of p.
i
p i (1 p ) i
1
That is P ( x1 , x2 ,..., xn / xi t )
n .
x
n
x
n
Ct
i
Ct p i (1 p ) i
x
n
x
Hence a statistic t is said to be sufficient for the parameter , if it contains all
information about the parameter contained in the sample, or
If the conditional distribution of any other statistic given t = r, is independent of .
Fisher-Neymaan Factorization Theorem (Condition for Sufficiency):
Let x1 , x2 ,..., xn be a random sample from a population with pmf/pdf f ( x, ) then
the joint pmf/pdf of the sample (usually called the likelihood of the sample) is
L( x1 , x2 ,..., xn , ) f ( x1 , ) f ( x2 , )..... f ( xn , ) , the statistic t is a sufficient estimator of , if
and only if it is possible to write
L( x1 , x2 ,..., xn , ) L1 (t , ) L2 ( x1 , x2 ,..., xn )
where L1 (t , ) is function of t and alone and L2 ( x1 , x2 ,..., xn ) is a function independent of
.
Proof:
If t is a sufficient estimator of , then the conditional distribution of x1 , x2 ,..., xn
given t = r is independent of . That is,
P ( x1 , x2 ,..., xn / t r ) h( x1 , x2 ,..., xn ) , which is independent of ------- (1)
Butbut P ( x1 , x2 ,..., xn / t r )
P( x1 , x2 ,..., xn ) L( x1 , x2 ,..., xn , )
P(t r )
P (t , )
Then by (1), for sufficient estimator t,
L( x1 , x2 ,..., xn , )
h( x1 , x2 ,..., xn )
P(t , )
L( x1 , x2 ,..., xn , ) h( x1 , x2 ,..., xn ) P (t , )
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Problem 1: For a Poisson distribution with parameter , show that sample mean x is the
sufficient estimator of .
Solution:
Let x1 , x2 ,..., xn be the random sample taken from P ( )
x
x1 x2 ... xn
n
L( x1 , x2 ,..., xn ) f ( x1 , ) f ( x2 , )... f ( xn , )
e 1 e 2
e xn
.
.....
x1 !
x2 !
xn !
x
x
e n i
x1 ! x2 !...xn !
x
e n
xi
e n nx
1
x1 ! x2 !...xn !
1
x1 ! x2 !...xn !
L1 ( x , ) L2 ( x1 , x2 ,..., xn )
1
x1 ! x2 !...xn !
where L1 ( x , ) e n nx and L2 ( x1 , x2 ,..., xn )
Hence, by factorization theorem, x is a sufficient estimator of .
Problem 2:
Let x1 , x2 ,..., xn be the random sample taken from a population with p.d.f.
f ( x, ) x 1 ; 0 x 1, 0 . Find a sufficient estimator for .
Solution:
Likelihood function L( x1 , x2 ,..., xn , ) f ( x1 , ) f ( x2 , )... f ( xn , )
x1 1 . x2 1 ...... xn 1
n
n
x
i 1
1
i
1
n
xi n
i 1
xi
n
i 1
n
L1 ( xi , ) L2 ( x1 , x2 ,..., xn )
i 1
n
where L1 ( xi , ) xi and L2 ( x1 , x2 ,..., xn )
i 1
i 1
n
1
n
n
x
i 1
STATISTICAL INFERENCE
, then by
i
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n
factorization theorem t xi can be considered as a sufficient estimator of .
i 1
Problem 3: Obtain a sufficient estimator for p, using samples x1 , x2 ,..., xn taken from B (n, p).
Solution:
The likelihood function of x1 , x2 ,..., xn ,
L( x1 , x2 ,..., xn , ) f ( x1 , p) f ( x2 , p)... f ( xn , p)
nCx1 p x1 (1 p)n x1 . nCx2 p x2 (1 p)n x2 ..... nCxn p xn (1 p)n xn
x
n x
p i (1 p) i nCx1 . nCx2 ... nCxn
L1 ( x , p ) L2 ( x1 , x2 ,..., xn )
where L1 ( x , p) p nx (1 p)nnx , and L2 ( x1 , x2 ,..., xn ) nCx1 . nCx2 ... nCxn , then by factorization
theorem x is a sufficient estimator of p.
Problem 4: If t is sufficient estimator for , prove that
log L
is a function of t and only.
Solution:
If t is a sufficient estimator of , then the likely hood function
L( x1 , x2 ,..., xn , ) L1 (t , ) L2 ( x1 , x2 ,..., xn )
log L( x1 , x2 ,..., xn , ) log L1 (t , ) log L2 ( x1 , x2 ,..., xn )
log L( x1 , x2 ,..., xn , )
log L1 (t , ) 0
Since, L1 (t , ) is a function of t and only,
log L is also a function of t and only.
Problem 5: If t is sufficient estimator for , then prove that any 1-1 function of t is also sufficient
for .
Solution:
Let h g (t ) , assume h is a 1-1 function of t, then t g 1 (h)
Since t is sufficient for ,
L( x1 , x2 ,..., xn , ) L1 (t , ) L2 ( x1 , x2 ,..., xn )
L ( x1 , x2 ,..., xn , ) L1 ( g 1 ( h), ) L2 ( x1 , x2 ,..., xn )
Hence L( x1 , x2 ,..., xn , ) is the product of a function of h and only and a function
independent of . Then by factorization theorem h is also sufficient for .
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2.4. Method of Estimation
(i)
Maximum Likelihood Estimator:
Let x1 , x2 ,..., xn be the sample taken from the population with p.m.f/p.d.f
f ( x, 1 , 2 ,.. k ) , where 1 , k ,.. k are the parameters involved. The likelihood function of
the sample L( x1 , x2 ,..xk , 1 , 2 ,.. k ) f ( x1 , 1 , k ,.. k ). f ( x2 ,1 , k ,.. k )..... f ( xn ,1 , k ,.. k ) .
The method of maximum likelihood suggests, the best estimators for estimating the
parameters 1 , k ,.. k are the estimators which maximizes the likelihood function. Such
estimators are known as Maximum Likelihood Estimators (M.L.E) of 1 , k ,.. k .
The Principle of M.L.E says that the best estimators of the parameters based on a
sample obtained are, those values of the parameters which make the probability of getting
that sample a maximum.
Using the method of differential calculus, the function of sample values for a
parameter which maximizing the likelihood function- called MLE of that parameter, can
be obtained. Let L( x1 , x2 ,..xn , 1 , 2 ,.. k ) be the likelihood function corresponds to the
sample x1 , x2 ,..xn . The value of 1 , as a function of x1 , x2 ,..., xn , maximizing the likelihood
2 L
L
0 . But since we
0 , and if for that value of 1 ,
12
1
know the value of 1 , which maximizing the likelihood function also maximizes logL, such
function can be obtained from
2 log L
log L
value of 1 can also be obtained by using
is less
0 if for that value of 1 ,
12
1
than zero.
Maximum Likelihood Estimators possess some desirable properties of a good
estimator.
(i)
MLE’s are asymptotically unbiased.
(ii)
MLE’s are consistent.
(iii)
MLE’s are most efficient.
(iv)
MLE’s are sufficient, if sufficient statistics exist.
(v)
MLE’s are asymptotically normally distributed.
Problem 1: Find the M.L.E. of and , using the random sample x1 , x2 ,..., xn taken from the
normal population N ( , )
Solution:
Given the random sample x1 , x2 ,..., xn from N ( , )
The likelihood function,
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1
2
L( x1 , x2 ,..xk , , )
x
1
2
e 2
2
1
2
x
2
2
e 2
1
2
n
2
xi 0
...
1
2
n xi
2 2
i 1
xi (1)
x 2
n
2
e 2
2
0
2 2
i 1
n
2
n x 2
i
2
e i 1 2
n
1
log L( x1 , x2 ,..xk , , ) n log
2
log L
0
1 n
xi x
n i 1
i 1
2 log L
1
2 ( n) 0
2
Hence, x is the MLE of
To obtain the MLE of ,
1
1 n x
n 2
2 i 3 (2) 0
2
2
i 1
2
log L
0
n
n
xi
3
i 1
n
x
2
0
n 2
1 n
2
xi
n i 1
n
xi
2 log L
n
3
2
2
4
i 1
At 2
2
i
i 1
2
2
1 n
2
xi ,
n i 1
2 log L
2
n2
n
x
i 1
i
n
x
n
x
i 1
2n 2
i 1
STATISTICAL INFERENCE
3
2
n2
2
i
0
2
i
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1 n
2
xi . Since MLE of is x , MLE of
n i 1
Then the MLE of 2 is
considered as
2 is
1 n
2
xi x , which is the sample variance.
n i 1
Problem 2: Find the MLE of , based on random samples taken from Poisson population with
parameter .
Solution:
Let x1 , x2 ,..., xn are the random sample taken from P ( ) , then
L( x1 , x2 ,..., xn , ) f ( x1 , ). f ( x2 , )... f ( xn , )
e n i
x1 ! x2 !...xn !
e 1 e 2
e xn
.
.....
x1 !
x2 !
xn !
x
x
log L n
x
x log log( x ! x !...x !)
i
1
2
n
i
log L
0 n
2 log L
2
at x ;
i
i
1
(0) 0
1
xi x
n i
2 log L
2
x
1
x
i
2
i
x
i
i
x
2
0 ( samples xi ' s from
Poisson population are 0 for P ( ))
Hence, x is the MLE of
Problem 3: Obtain the MLE of for the following distribution f ( x)
1 | x |
e
, x .
2
Solution:
Let x1 , x2 ,..., xn be the sample from the given population, then,
L( x1 , x2 ,..., xn , ) f ( x1 , ). f ( x2 , )... f ( xn , )
1 | x1 | 1 | x2 | 1 | xn |
e
. e
.... e
2
2
2
n
1
e
2
STATISTICAL INFERENCE
| xi |
i
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School of Distance Education
Therefore, log L n log 2 | xi |
i
Here log L is maximum when
| x | is minimum.
i
i
This happens when is the median of the random sample x1 , x2 ,..., xn . So MLE of
is the median of x1 , x2 ,..., xn .
Problem 4: Obtain the MLE of and using the random samples x1 , x2 ,..., xn taken from the
x
1
population with pdf f ( x) e
, x , 0 .
Solution:
The likelihood function L (x1, x2, . . . , xn, , ) can be written as
1
L( x1 , x2 ,..., xn , , ) e
x1
n
1
e
x2
1
. e
1
.... e
i
(x )
i
i
log L
n
0 0
0
log L
n
0
i
i
2
i
(1)
0
n
i
1
n
------
(x )
(x )
xi
log L n log
xn
(x )
i
-----
(2)
i
Equation (1) cannot imply the MLE of . But we know log L is maximized when
( xi ) a minimum is. This happens when is a maximum. But cannot be greater
i
than Min xi . Hence Min xi is the MLE of . Then by (2) , the value of can be written as
1
( xi Min xi ) and it can be verified that atthis value of ,
n i
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2 log L
1
0 . Hence the MLE of is
2
n
( x Min x ) .
i
i
i
Problem 5: Obtain the MLE of a and b using the random samples x1 , x2 ,..., xn taken from a
rectangular population over the interval ( a b , a b ) .
Solution:
Here random samples x1 , x2 ,..., xn taken from a rectangular population over the
interval ( a b , a b ) . Hence f(x) is given by
f ( x)
1
1
, a b x ab
a b a b 2b
In this case the likelihood function
1
L( x1 , x2 ,..., xn , a, b)
a b a b
n
The method of differential calculus cannot be applied here. The likelihood function L is
maximum when a b a b is minimum. This happens when a b is taking its
minimum and a b is taking its maximum possible value.
But a b cannot be less than the largest value of x1 , x2 ,..., xn
and
a b
cannot be
greater than the smallest value of x1 , x2 ,..., xn .
Hence MLE of a b = Max ( x1 , x2 ,..., xn ) and
MLE of a b = Min ( x1 , x2 ,..., xn )
Then,
the MLE of a
Max( x1 , x2 ,..., xn ) Min( x1 , x2 ,..., xn )
and
2
the MLE of b
Max ( x1 , x2 ,..., xn ) Min( x1 , x2 ,..., xn )
2
Problem 6: Obtain the MLE of the parameter using the random samples x1 , x2 ,..., xn taken
1
from a population with pdf, f ( x ) , 0 x .
n
The likelihood function
1
1
L( x1 , x2 ,..., xn , ) = n
The likelihood function getting its maximum value when n is minimum, ie., is
minimum. But for the given pdf, 0 x , so, cannot be less than the highest
observation of the sample . Hence the minimum possible value of is the highest value
of the sample. That is the MLE of = Max ( x1 , x2 ,..., xn )
Problem 7: If a sufficient statistics T exists for , then prove that MLE of is a function of T.
Solution:
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If T is sufficient for estimating , then,
L( x1 , x2 ,..., xn , ) L1 (T , ) L2 ( x1 , x2 ,..., xn )
log L
0
L1 (T , ) 0
log L
log L1 (T , ) 0 and
This implies is a function of T.
e x
, where X can assume only
x!
nonnegative integral values and given the following observed values, 4,5,7,21,24,12,15,7,9,14.
Find the M.L.E of .
Problem 8: Given that the frequency function f ( x, )
Solution:
Here 10 observations are taken from the given Poisson population. If let
x1 , x2 ,..., x10 are the random sample taken from P ( ) , then
L( x1 , x2 ,..., x10 , ) f ( x1 , ). f ( x2 , )... f ( x10 , )
e 1 e 2
e 10
.
.....
x1 !
x2 !
x10 !
x
x
x
e10 i
x1 ! x2 !...x10 !
x
x log log( x ! x !...x
log L 10
i
1
2
10
!)
i
log L
0 10
at x ;
i
i
1
(0) 0
1
xi x
10 i
2 log L
2
2 log L
2
x
1
x
i
2
i
x
i
i
x
2
0 ( samples xi ' s from
Poisson population are 0 for P ( ))
Hence,
1
xi is the MLE of .
10 i
1
4 5 7 21 24 12 15 7 9 14
= 11.8.
xi =
10
10 i
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(ii)
Method of moments:
Let f ( x, 1 , 2 ,..., n ) be the pdf of the population and x1 , x2 ,..., xn be the random
sample taken from it. In the method of moment, we find the first k moments of the
population and equate them to the corresponding moments of the sample. The values of
1 , 2 ,..., n , obtained as a function of x1 , x2 ,..., xn , by solving the equations are considered as
the moment estimators of 1 , 2 ,..., n .
Problem 1: For a normal population N ( , ) , find the estimators of and by the method of
moments.
Solution:
Let x1 , x2 ,..., xn be the random sample taken from N ( , ) ,
First raw moment of the population E ( X ) .
First raw moment of the sample is x .
of .
Equating first moment of the sample and the population we get x as the estimator
Second raw moment of the population E ( X 2 ) E ( X ) V ( X ) 2 2
2
Second raw moment of the sample is
1 n 2
xi
n i 1
1 n 2
xi
n i 1
Equating these two, we get 2 2
Hence moment estimator of 2 is
2
1 n 2
xi 2 ;
n i 1
But x
is the estimator of .
1 n 2
xi x 2 , which is the sample variance.
n i 1
Problem 2: X is a random variable with probability masses as shown below.
X:
0
f ( x) : 1 2
1
2
2
;
0 1 , Find the
Moment estimate of , if in a 25 observations there were 10 ones and 4 twos.
Solution:
Out of 25 samples taken from the population, it is recorded 10 ones, 4 twos and
the remaining 9 zeroes.
Then first moment of the sample is
9 0 10 1 4 2 18
,
25
25
First moment of the population,
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E ( X ) 0 1 2 1 2 2 2 2
Equating these we get,
2 2
18
,
25
ie., 50 2 25 18 0
Solving this quadratic expression, we get = 0.295.
Problem.3: Obtain the moment estimate of , if
X:
1
2
3
4
0 1,
1
1
1
1 ;
f ( x) :
4
4
4
4
1,5,7 and 7 respectively.
the
probability
masses
are
and the observed frequencies are
Solution:
Out of 20 samples taken from the population, it is recorded 1 one, 5 twos, 7
threes and the remaining 7 fours.
Then first moment of the sample is
1 1 5 2 7 3 7 4 60
= 3,
20
20
First moment of the population,
1
1
1
1
E ( X ) 1
2
3
4
4
4
4
4
10 4
4
4
Equating these we get,
10 4
3,
4
4
solving this quadratic expression, we get = 0.5 .
EXERCISES
1. X 1 , X 2 , X 3 are random samples from population with mean and standard deviation
. T1 , T2 , T3 are defined as
T1 X 1 X 2 X 3 ; T2 2 X 1 3 X 3 4 X 2 ; and
1
T3 X 1 X 2 X 3 . Are (i) T1 and T2 are unbiased estimator? (ii) Find , such that
9
T3 is an unbiased estimator of (iii) Which is the most efficient estimator?
2. Define consistent estimator. Obtain the sufficient conditions for consistency.
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3. For Poisson distribution with parameter , show that
1
is a consistent estimator of
x
1
.
4. Let x1 , x2 ,..., xn are random sample taken from N ( , ) . Find sufficient estimators of
and .
5. Define sufficient statistic. Find a sufficient statistic when f ( x, )
1
, 0 x
6. Find the MLE of p, based on sample taken from Binomial distribution with parameters
N and p.
7. Obtain the MLE of in f ( x, ) (1 ) x , 0 x 1 based on random sample
x1 , x2 ,..., xn taken from the population. Also verify whether the MLE is a sufficient
estimator of .
8. Find the MLE of , where the random samples x1 , x2 ,..., xn are taken from the
1 x
population with pdf f ( x) e , 0 x
9. An urn contains white and black balls in unknown proportions, the total number of
balls being 12. Four balls are drawn at random, of which 3 are found to be white and 1
black. Find the maximum likelihood estimate of the number of white balls in the urn.
10. Explain the method of maximum likelihood estimation. Find M.L.E. of , when
1
1
x
2
2
0, otherwise
f ( x, ) 1, if
11. Find the moment estimator of , if
1
x p 1e
x
, 0 x ; p is known
p p
based on the random samples x1 , x2 ,..., xn are taken from the population.
f ( x, )
12. Explain the method of moments. Using this, obtain the estimators of the parameters a
and b of a uniform distribution over the interval [a,b]
13. Find an estimator of , based on random samples taken from Poisson population with
parameter by the method of moments.
14. Obtain
X:
the moment estimate of ,
1
2
3
4
1
1
1
1 ;
f ( x) :
4
4
4
4
frequencies are 1,5,7 and 7 respectively.
if
the
probability
0 1,
masses
are
and the observed
********************
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CHAPTER 3
INTERVAL ESTIMATION
3.1. Interval Estimation
In point estimation we are finding an estimator to estimate the unknown parameter
under the expectation that the true value of the parameter is very close to the value of the
estimator suggested. We consider the value of the estimator as the value of the parameter.
For eg., in case of N ( , ) , x is suggested as an estimator of . But when we are dealing
with interval estimation, we are estimating an interval where the value of the unknown
parameter lying with a pre-assigned probability.
That is in case of interval estimation for a parameter , it is estimating two statistics
t1 and t2 (t1 t2 ) such that the probability that the interval ( t1 , t2 ) contains the true value
of the unknown parameter with a specified probability 1 ; 0 1 . Then ( t1 , t2 ) is
termed as 100 1 % confidence interval for the parameter , and
1
is the
confidence coefficient. It is to be noted that there may be many confidence interval for a
particular parameter with same confidence coefficient. Shortness, stability etc., are
some desirable property to identify a good interval.
3.2. Confidence interval for the mean of a normal population with confidence
coefficient 1 :
Case I: When is known
Let x1 , x2 ,..., xn be the sample taken from N ( , ) and let the sample mean be x . We
use x - the point estimator of for its interval estimation. The mean x follows N ( ,
or
t
x
n
),
n
~ N ( 0,1 ) .
From standard normal table it can observe the value t such that,
2
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P ( | t | t ) 1
2
P( |
x
P ( t
| t ) 1
2
x
n
2
P ( x t
2
Multiplying by -1
n
t ) 1
2
x t
) 1
n
n
2
P ( x t
2
P ( x t
2
n
n
x t
2
x t
2
) 1
n
) 1
n
Hence the confidence interval for with confidence coefficient 1 is,
, x t
x t
.
n
n
2
2
Case II: When is un-known
Let x1 , x2 ,..., xn be the sample taken from N ( , ) and let the sample mean be x . It
is already derived for random samples taken from N ( , ) , t
x
s
n 1
~ tn1 .
From the table of t-distribution, it can be observed a number t such that,
2
P( | tn 1 | t ) 1
2
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P( |
x
s
n 1
| t ) 1
2
s
x t
n 1
2
P ( t
2
P ( x t
2
P ( x t
2
s
) 1
n 1
s
x t
n 1
2
s
) 1
n 1
s
x t
n 1
2
s
) 1
n 1
Hence the confidence interval for with confidence coefficient 1 is,
x t
2
s
, x t
n 1
2
s
.
n 1
Problem 1: Estimate a 95% confidence interval for , based on 10 random samples
17,21,20,18,19,22,20,21,16,19 taken from N ( ,3)
Solution:
Here is known. Hence the 100 1 % confidence interval for is ,
, x t
x t
n
n
2
2
We have, x
17 21 20 18 19 22 20 21 16, 19
19.3
10
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= 3 ; Given 1 = 0.95,
Hence,
from standard normal table we get, t 1.96 ,
2
so that,
P ( | t | t ) 0.95 .
2
Hence the confidence interval is,
3
3
, 19.3 1.96
19.3 1.96
= 17.44 , 21.16
10
10
Problem 2: Find the least sample size required if the length of 95% confidence interval for the
mean of a normal population with standard deviation 4 should be less than 5.
Solution:
Let
n random samples are taken from the population N ( , 4) . The confidence
interval for the mean is,
, x t
x t
n
n
2
2
Given the confidence coefficient is 95 %. Hence from the standard normal table t =
2
1.96.
To find the minimum number of samples such that, the length of the interval of ,
2
x t
5
x t
t
n
n
n
2
2
2
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2 4
5
n
2 4
1.96
n
5
1.96
2 4
n 1.96
5
2
n 9.83
Hence the least number of samples required is 10
Problem 3: A sample of size 17 taken from N ( , ) . Mean of the sample is 12 and the sample
variance is 4. Using the data, find a 90% confidence interval for .
Solution:
Here the value of is unknown. Then the confidence interval for is ,
x t
2
s
, x t
n 1
2
s
n 1
Since the confidence coefficient is 90%, from the table of t distribution for 16 d.f., we
get t = 1.746, so as P( | t16 | t ) 0.90 .
2
2
Also given x 12 and s
4 2
Then the 90% confidence interval for is,
2
2
, 12 1.746
12 1.746
11.127 , 12.873
16
16
Problem 4: For a N ( ,3) population, construct a 95% confidence interval for 3 5 , on the basis
of the random sample of size 25. The sample mean was found to be 30.
Solution:
Consider Y 3 X 5 ; then Y follows N (3 5, 32 2 ) N (3 5, 9 9)
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Sample mean corresponding to Y , y ~ N (3 5,
9
)
n
Here n=25 and, y 3 30 5 95 ( x 30)
u
1
Given
= 0.95,
y (3 5)
~ N (0,1)
9
25
hence from standard normal table ,
t 1.96 ,so as
2
P ( | t | t ) 0.95 . Hence, P ( |
2
y (3 5)
| 1.96 ) 0.95
9
25
This implies 95% confidence interval for 3 5 as
9
9
, y 1.96
y 1.96
25
25
Since y 95 ,the interval is
9
9
,95 1.96
95 1.96
91.472, 98.528
25
25
Problem 5: Show that the length of the confidence interval for the mean of a normal population
with known variance can be made however small we please by increasing the sample size.
Solution:
The confidence interval for the mean of a normal population when 2 is known is
given by
, x t
x t
.
n
n
2
2
The length of the interval = x t
x t
n
n
2
2
2 t
2
STATISTICAL INFERENCE
,
n
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Here as the number of sample n increases the length of the interval decreases and
which can be adjusted to any given number for a given significance level by selecting
suitable number of samples.
3.3. Confidence interval for the difference of means of two normal populations having
known common variance 2 :
Let n1 and n2 are the number of samples independently drawn from to normal
populations N 1 , and N 2 , respectively. Let x1 and x2 be the means and S1 and
S 2 be the standard deviations of the samples drawn from the first and second population
respectively.
x1 ~ N 1 ,
and x2 ~ N 2 ,
n1
n2
Then,
2 2
x1 x2 ~ N 1 2 ,
n
n
1
1
t
x1 x2 1 2
1 1
n1 n1
~ N (0,1)
From standard normal table it can observe the value t such that,
2
P ( | t | t ) 1
2
P( |
x1 x2 1 2
1 1
n1 n1
| t ) 1
2
1 1
1 1
P x1 x2 t
1 2 x1 x2 t
n1 n1
n1 n1
2
2
1 1
P x1 x2 t
n1 n1
2
STATISTICAL INFERENCE
1 2
x1 x2 t
2
1 1
n1 n1
1
1
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Hence the 100 (1 ) % confidence interval for 1 2 is,
1 1
,
x1 x2 t
n1 n1
2
x1 x2 t
2
1 1
n1 n1
Problem 1: The average mark scored by 32 boys in an examination is 72 with a standard deviation
of 8, while that scored by 32 girls is 70 with a standard deviation of 6. Construct a 99 %
confidence interval for the difference of means. (Assume S.D’s are equal)
Solution:
The confidence interval for difference in mean,
1 1
,
x1 x2 t
n1 n1
2
The common value for variance
=
x1 72 ; 1 8; n1 32
x1 x2 t
2
1 1
n1 n1
n1s12 n2 s2 2
(since we have large samples)
n1 n2
32 82 32 62
32 32
= 7.07.
and x2 70 ; 2 6; n2 32
Confidence coefficient is 99%. From standard normal table t 2.57
2
Hence the confidence interval is,
1
1
,
72 70 2.57 7.07
32 32
72 70 2.57 7.07
1
1
32 32
= 2 4.543, 2 4.543
= 2.543, 6.543 .
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3.4. Confidence interval for the variance of a normal population:
Let x1 , x2 ,..., xn be the sample taken from N ( , ) with sample variance S 2 . Then,
2
nS 2
follow chi-square distribution with (n-1) d.f.
2
From the table of 2 -distribution, identify the numbers 2
and 2
2
P 2 ( n 1) 2
2
2
and P 2 ( n 1) 2
1
2
1
P 2 2 2
1
2
2
nS 2
ie., P 2
2
2
1
2
2
1
nS 2
nS 2
P 2
2 2
1
2
2
nS 2
nS 2
P 2 2 2
1
2
2
2
so as,
respectively.
1
2
1
2
2
1
2 1
2
P
2
2
nS 2
nS
1
1
1
Hence the confidence interval for 2 with confidence coefficient 1 is,
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nS 2
2
2
2
nS
2
,
1
2
Problem 1: A sample of size 12 taken from N ( , ) . Mean of the sample is 10 and the sample
variance is 9. Find a 90% confidence interval for 2 .
Solution:
Given n = 12, S 2 = 9
nS 2
Confidence interval for 2 is given by 2
2
2
,
nS
2
1
2
For 90% confidence interval, ie., for 0.10 , from table of chi-square distribution for 11
d.f,
P 211 4.58
0.10
2
1
2
1
2
0.95 and P 211 19.68
2
4.58 and
0.10
2
0.05
19.68
2
12 9
Hence the 90% confidence interval for 2 is,
19.68
5.49 , 23.58
,
12 9
4.58
Problem 2: An optical firm purchases glass for making lenses. Assume that the refractive index of
20 pieces of glass have variance of 1.20 X 104 . Construct a 95% confidence interval for the
population variance.
Solution:
Confidence interval for 2 with confidence coefficient 1 is,
nS 2
2
2
STATISTICAL INFERENCE
2
,
nS
2
1
2
.
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Here 20 samples are drawn and the sample variance is 1.20 X 104 . For confidence
coefficient 95%, and for (20 – 1 ) = 19 d.f. , chi-square table implies,
2 = 32.8523 and 2
1
2
2
Hence the 95% confidence interval for 2 is,
= 8.9066.
20 1.20 104
32.8523
,
20 1.20 104
8.9066
= 0.7305 10 4 , 2.6946 10 4 .
Problem 3: Construct a 95% confidence interval for the variance 2 of the normal population
with unknown mean using the following sample:
4.5, 10.2, 10.5, 9.8, 13.0, 19.2, 15.5, 13.3, 10.8 and 16.4
Solution:
Given n = 10 samples from the normal distribution.
nS 2
(1 )% Confidence interval for 2 is given by 2
2
2
,
nS
2
1
2
For (1 )% =95% , 0.05 ; from table of chi-square distribution for n-1 = 9, d.f,
P 29 2.7004
2
1
1
2
0.05
2
0.975 and P 29 19.0228
2.7004 and
2
0.05
2
0.025
19.0228
2
To find the sample variance of the population, using the given 10 samples
The calculations follow:
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x
x2
4.5
20.25
10.2
104.04
10.5
110.25
9.8
96.04
13.0
169
19.2
368.64
15.5
240.25
13.3
176.89
10.8
116.64
16.4
268.96
123.2
1670.96
x
s
123.2
12.32
10
1
2
x12 x
10 i
1
2
1670.96 12.32 3.9
10
10 3.9 2
Hence the 90% confidence interval for is,
19.0228
2
STATISTICAL INFERENCE
,
10 3.9
2.7004
2
7.996 , 56.325 .
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3.5. Confidence interval for large samples:
1. 100 1 % Confidence interval for the proportion of a binomial population:
Consider a binomial population with parameters N and p. Assume N is known.
Repeat the Bernoulli trial for n number of times. Then the binomial variable X, follow
B(n,p). When n becomes very large, X follows normal distribution N (np, npq ) .
Then, t
X np
~ N (0,1)
npq
X
p
t n
~ N (0,1)
pq
n
As an approximation,
t
(where, p
p p
~ N (0,1) ;
pq
n
q 1 p
X
is the sample proportion)
n
Hence from standard normal table, obtain t such that,
2
P ( | t | t ) 1
2
P( |
p p
| t ) 1
pq
2
n
STATISTICAL INFERENCE
P ( t
2
p p
t ) 1
pq
2
n
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pq
p p t
n
2
P ( p t
2
P ( p t
2
ie., P ( p t
2
pq
p p t
n
2
pq
p p t
n
2
pq
) 1
n
pq
) 1
n
pq
) 1
n
Hence the 100 1 % confidence interval for p for large n is,
p t
2
pq
, p t
n
2
pq
n
where,
q 1 p
Problem 1: Random samples of 120 workers of a factory 40 are dissatisfied with their working
conditions. Form a 95% confidence interval for the proportion of dissatisfied workers of the factory.
Solution:
Given
the sample proportion of dissatisfied workers p =
40
1
. Also given the
120 3
confidence coefficient = 95%
The confidence interval for proportion of dissatisfied workers is given by,
p t
2
pq
, p t
n
2
pq
n
For 95% confidence coefficient, from table of standard normal distribution, t 1.96
2
Hence the confidence interval is,
1 2
1 2
1
1
3
3 0.214 , 0.4526
1.96 3 3 ,
1.96
120
3
120
3
Problem 2: Each computer chip produced by a certain manufacturer is either acceptable or
unacceptable. A large batch of such chips produced and it is supposed that each chip in this batch
will be independently acceptable with some unknown probability p. To obtain a 99% confidence
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interval for p, which is to be of length approximately 0.05, a sample of size 30 is initially taken. If
24 of the 30 chips are deemed acceptable, find the approximate sample size.
(n=1492)
Solution:
The confidence interval for proportion of mortality rate is given by,
p t
2
pq
, p t
n
2
pq
n
The given confidence coefficient is 99%. From standard normal table t 2.57
2
From the 30 samples taken, p
24 4
.
30 5
The length of the confidence interval = 2t
2
pq
.
n
To find the approximate sample size so as length of 99% confidence interval
2t
2
pq
= 0.05 .
n
That is to get n, such that
4 4
1
5 5
2 2.57
0.05
n
2 2.57 4 4
n
1 1690.85
0.05 5 5
2
1691
2. 100 1 % Confidence interval for in Poisson population:
Consider n random samples from Poisson population and assume
n is large. As n become very large, the Poisson random variable X follows
normal distribution N ( , ) .
The sample mean x ~ N ( ,
t
) for large n.
n
x
~ N (0,1), approximately ( E ( x ) )
x
n
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From standard normal table, obtain t such that,
2
P ( | t | t ) 1
2
x
x t
n
2
P ( t
2
x
x t
n
2
P ( x t
2
P ( x t
2
x
) 1
n
x
x t
n
2
x
) 1
n
x
) 1
n
Hence the confidence interval with confidence coefficient 1 is,
x t
2
STATISTICAL INFERENCE
x
, x t
n
2
x
n
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EXERCISES
1. The mean of a sample of size 30 drawn from a normal with mean and variance
36 was recorded as 45.6. Find a 95% confidence interval for .
2. Construct a 100 1 % confidence interval for the difference of means of two
normal populations having common variance.
3. Obtain 100 1 % confidence interval for variance 2 of a normal population
N ( , ) , where the mean is known.
4. Derive 95% confidence interval for the mean of a normal population when (i)
variance is known (ii) variance is unknown.
5. Obtain 95% confidence interval for the parameter of Poisson distribution on the
basis of a random sample of size n.
6. A random sample of size 20 is drawn from a normal population N ( , ) .
Sample
mean and the sample variance are respectively 22 and 16. Find a 90% confidence
interval for .
7. The mean of a sample of size 24 drawn from N ( , ) is 25.5. The sample variance
is 14. Construct a 95% confidence interval for 2 .
8. Using a random sample of size 40 drawn from a Poisson population, construct a
90% confidence interval for the parameter .
9. In a random sample of 40 articles 40 are found to be defective.
Obtain 95%
confidence interval for the true proportion of defectives in the population of
articles.
10. Obtain a large sample 100 (1 ) % confidence interval for the parameter , in
random sampling from the population with pdf f ( x) e x , x 0 , 0 .
*********************
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CHAPTER 4
TESTING OF HYPOTHESIS
4.1. Statistical Hypothesis:
Statistical inference mainly deals with estimation and testing of hypothesis. As we
already seen, the area of estimation consists of making an estimate of an appropriate value
of the unknown parameter or an interval for the unknown value of the parameter with a
specified probability.
A statistical hypothesis is some statement or assertion about the population
parameters or about the form of probability distribution or the population.
Theory on
testing of hypothesis was initiated by J.Neymaan and E.S.Pearson.
Simple and composite hypothesis:
A statistical hypothesis which completely specifies the population, ie., it specifies
the values of all parameters involved in the probability distribution of the population,
then it is called simple hypothesis, otherwise it is called composite.
For eg., assume x1 , x2 ,..., xn are n random samples taken from a normal population
N ( , ) .
Then the hypothesis H : 1 , 1 is a simple hypothesis. But the
hypotheses (i) H : 0 ; (ii) H : 1 ; (iii) H : 0 , 1
(iv)
H : 0 , 1 (v) H : 0 , 1 etc., are not specifying on the exact values of all
the parameters involved, hence they are considered as composite hypothesis.
4.2. Testing of Hypothesis:
Testing of hypothesis is a decision making whether to accept or reject the proposed
hypothesis about the population based on the random samples drawn from the
population.
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Consider the situation where a light bulb manufacturing firm introducing a new
type of light bulb. They are claiming that the new product is superior to the existing
standard type in terms of the life length. Suppose to test the claim of the firm. Assume it
is known that the average life length of the bulb of standard type as 500 hrs.
Here we
propose a hypothesis regarding the average life length , of the new type of light bulb as
H : 500 . This hypothesis is to be tested, against the alternatives, (i) is greater than
500 or (ii) is less than 500 or (iii) either >500 or <500; ie., 500 .
In a statistical testing of hypothesis, the hypothesis is to be tested is termed as Null
hypothesis.
The null hypothesis is denoted by H 0 .
Here the null hypothesis is
H 0 : 500 .
Another hypothesis in our mind which we will accept or reject according as we
reject or accept H 0 is termed as Alternative hypothesis, denoted by H1 .
In the above
illustration, H1 can be (i) H1 : 500 or (ii) H1 : 500 or (iii) H1 : 500 .
In this situation we are taking a random sample of n bulbs of new type and find the
average life length of the sample item. Based on the sample mean (which is a good
estimator of population mean) we decide whether to accept or reject the null hypothesis.
Roughly speaking, if the alternate hypothesis considered is H1 : 500 , and the sample
mean is much higher than 500, the hypothesis H 0 : 500 is rejected. If H1 : 500 , and
the sample mean is much lesser than 500, H 0 is rejected and if H1 : 500 , and the sample
mean is reasonably distant from 500, H 0 is rejected.
Here we are making decision based on the sample mean, because we had to make
a decision on population mean and sample mean is a good statistic to say something
about the population mean. As this, in any statistical test we have to find an appropriate
statistic to make decision based on its value. The value of statistic can be calculated by the
value of the samples selected. Such a statistic used in testing of hypothesis is termed as
test statistic.
In a statistical test, according to the alternative hypothesis selected, we divide the
range of variation of the test statistic in to two. One is acceptance region and the other is
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rejection region. If the value of the test statistic is in the rejection region, H 0 is rejected.
The rejection region is known as critical region.
4.3. Errors in Testing of Hypothesis:
In a hypothesis testing procedure, since decision is making is based on the sample
drawn from the population, there exists possibilities for two types of errors. They are
termed as Type-I error and Type-II error.
Type-I error: It is an error due to rejecting the null hypothesis H 0 , when H 0 is true.
Type-II error: It is an error due to accepting the null hypothesis H 0 , when H 0 is false.
The possible errors can be tabulated as follows:
Action taken
H 0 is
H 0 is
Based on the
true
false
Reject
TYPE-I
NO
H0
ERROR
ERROR
Accept
NO
TYPE-II
H0
ERROR
ERROR
sample
Using better
statistical criteria, the possible errors in testing procedure can be minimized.
4.4. Steps in Testing of Hypothesis:
(i)
Define the population and formulate the hypothesis.
(ii)
Choose an appropriate test statistic.
(iii)
Divide the range of variation of the test statistics into two regions, acceptance
region and rejection region, considering some probabilistic restrictions.
(iv)
Take a random sample from the given population and calculated the value of
the test statistic and decide whether to accept or reject the hypothesis.
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In a testing procedure, the null hypothesis is rejected when the value of the test
statistic falls in the pre-decided rejection region. Since the value of the test statistic is
decided by the sample drawn, there is a chance for the value of the test statistic to fall in
the rejection region even though the null hypothesis is true (type-I-error). The probability
of the value of the test statistic to fall in the rejection region, even though the null
hypothesis is true is known as significance level or size of the test, denoted by .
ie.,
Significance level = P (Rej. H 0 / H 0 is true) = P (Type-I-error)
The probability of the value of the test statistic to fall in the rejection region, when
the alternative hypothesis is true is known as
ie.,
power of the test,
denoted by .
Power of the test = P (Rej. H 0 / H1 is true) = 1 - P (Acc. H 0 / H1 is true)
= 1 - P (Acc. H 0 / H 0 is false) = 1 - P (Type-II error)
Problem 1: To test H 0 : 1 against the alternative H1 : 2 , based on a random sample of size
one taken from the population with pdf
1
, 0 x
. Find the size and power of the tests if the
0, otherwise
f ( x, )
critical regions are (i) x>0.5 (ii) 1 < x < 1.5
Solution:
Let x be the random sample drawn from the population.
Given H 0 : 1 and H1 : 2
(i) When the test is with critical region x > 0.5,
Size of the test = P( Rej H 0 / H 0 is true)
= P(x > 0.5 / 1 )
1
dx/ 1
0.5
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1
1 dx
0.5
x 0.5 1 0.5 0.5
1
Power of the test (1 ) = P (Rej. H 0 / H1 is true)
= P(x > 0.5/ 2 )
1
dx
/ 2
0.5
2
1
2 dx
0.5
1 2
x 0.75
2 0.5
(ii) When the test is with critical region 1 < x < 1.5,
Size of the test = P (Rej H 0 / H 0 is true)
= P (1 < x < 1.5 / 1 )
1.5
1
dx/ 1
, for the given population 0 x
1
. Hence when 1 , the critical region not exists.
0.
Power of the test (1 ) = P (Rej. H 0 / H1 is true)
= P (1 < x < 1.5 / 2 )
1.5
1
1.5
1
1
dx
dx = 0. 25
/
2
2
1
Problem 2: In a coin tossing experiment, let p be the probability of getting a head. The coin is
tossed 10 times to test the hypothesis H 0 : p 0.5 against the alternative H1 : p 0.7 . Reject H 0 , if
6 or more tosses out of 10 result in head. Find significance level and power of the test.
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Solution:
To test, H 0 : p 0.5 against H1 : p 0.7 . 10 tosses of the coin is considered and let
X denote total number of heads obtained. Then X follows binomial distribution B(10,p).
The critical region is X 6 .
Significance level = P (Type –I error)
= P (Rej H 0 / H 0 is true)
= P ( X 6 / p = 0.5)
10
10C x p x q10 x / p 0.5
x 6
10
10C x 0.5 x (1 0.5)10 x
x 6
386
210
Power of the test (1 - ) = P (Rej H 0 / H1 is true)
10
10C x p x q10 x / p 0.7
x 6
10
10C x 0.7 x (0.3)10 x 0.8495
x 6
Problem 3: Based on a single observation x, taken from an exponential population
f ( x) e x , x 0, 0 , to test H 0 : 2 against H1 : 1 . If the critical region suggested is,
x 1, find the probability of type–I and type- II errors. Also find the power function of the test, if
H1 suggested is H1 : r , where r 2 .
Solution:
To test, H 0 : 2 against H1 : 1 . Critical region is x 1.
is the parameter of exponential population then, f ( x) e x , x 0, 0 .
P(Type-I Error) = P( Rej H 0 / H 0 is true) = P( x 1/ 2 )
e x dx / 2
1
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2e
2 x
1
e 2 x
2
dx 2
e
2 1
Power of the test (1 ) = P (Rej H 0 / H1 is true)
= P ( x 1/ 1 )
e x dx / 1
1
e x
1
e dx
e
1 1
1
x
P (Type II error) = 1 – Power of the test = 1 e 1
When the alternative hypothesis is H1 : r , where r 2 ,
Power of the test (1 ) = P ( x 1/ r ), r 2
e x dx / r
1
re
1
rx
e rx
r
dx r
e ,r2
r 1
Power function (1 ) (r) = e r , r 2
Hence for different values of r , r 2 in alternative hypothesis the power of
the given critical region can be calculated.
Problem 4: In a city the milk consumption of families, x, is assumed following the distribution
with, p.d.f. f ( x)
1 x
e , x 0, 0 . To test H 0 : 5 against H1 : 10 . H 0 is rejected ,if a
family selected at random consumes 15 units or more. Obtain the size and power of the test.
Solution:
To test H 0 : 5 against H1 : 10 , reject H 0 , if x 15 .
Size of the test = P (Rej H 0 / H 0 is true) = P ( x 15 / 5 )
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x
1
15 e dx / 5
5x
e e 3
15
x
1
e 5 dx
5 15
Power of the test (1 ) = P ( x 15 / 10 )
x
1
e dx / 10
15
x
e 10
15
x
1
e 10 dx
10 15
e
15
10
e
3
2
.
Problem 5: x1 , x2 ,..., x9 are 9 random samples drawn from a normal population N ( ,5) . To test
H 0 : 5 against H1 : 8 . The critical region suggested is x 7 , where x the sample
is mean. Find significance level and power of the test
Solution:
To test H 0 : 5 against H1 : 8 . Critical region is x 7 .
Significance level = P (Type –I error)
= P (Rej H 0 / H 0 is true) = P ( x 7 / = 5)
Since x1 , x2 ,..., x9 are random sample from normal population, the sample mean x
is distributed as N ( ,
5
).
9
Hence U
x
5
9
~ N (0,1)
x 9 7 9
P( x 7 / = 5) = P
/ 5
5
5
STATISTICAL INFERENCE
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= P U 1.2 = 0.1151 (from std. normal table)
= P ( Rej H 0 / H1 is true)
Power of the test
= P ( x 7 / = 8)
x 9 7 9
= P
/ 8
5
5
P U 0.6 = 0.7257 (from std. normal table).
Problem 6:
f ( x)
x1 , x2
are 2 random
1 x
e , x 0 , 0 .
samples drawn
from
a population
with p.d.f.
To test H 0 : 2 against H1 : 4 . Reject the hypothesis if
x1 x2 9.5 . Obtain significance level and power of the test
Solution:
To test H 0 : 2 against H1 : 4 . Critical region is x1 x2 9.5 .
Significance level = P (Rej H 0 / H 0 is true) = P ( x1 x2 9.5 / 2 )
Here the given critical region is a plane in the first quadrant as given in the graph,
where x1 x2 9.5 , x1 0, x2 0 .
The joint density of x1 , x2 is given by
x
1 1 1
f ( x1 , x2 ) e e
STATISTICAL INFERENCE
x2
,x
1
0, x2 0 ( x1 , x2 are ind .)
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1
f ( x1 , x2 ) 2 e
x1 x2
, x1 0, x2 0
P ( x1 x2 9.5 / 2 ) = 1 - P ( x1 x2 9.5 / 2 )
Note that,
9.5 9.5 x1
1
e
2
P( x1 x2 9.5 / 2 ) =
x1 0 x2 0
9.5 9.5 x1
x1 0
1
16
1
e
16
x2 0
9.5 9.5 x1
0
1
4
9.5
1
4
9.5
e
dx2 dx1 / 2
x1 x2
4
dx2 dx1
x1 x2
4
dx2 dx1
0
e
x 9.5 x1
1
2
0
e
e
x
2
2
dx2 dx1
0
x
1
2
0
1
2
x1 x2
9.5
e
9.5 x1
0
2
2 e
e
dx1
9.5
2
e
x
1
2
dx1
0
1 9.5
x1e 2 2e
2
x1
2
9.5
0
9.5
9.5
9.5
1
2
9.5e
2e 2 2
2
0
1 4.75e 4.75 e 4.75
Hence, significance level = 1 - ( 1 4.75e 4.75 e 4.75 )
= 5.75e 4.75 = (0.05 approx.)
Power of the test
STATISTICAL INFERENCE
= P (Rej H 0 / H1 is true)
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= P ( x1 x2 9.5 / 4 )
= 1 - P ( x1 x2 9.5 / 4 )
9.5 9.5 x1
P ( x1 x2 9.5 / 4 ) =
x1 0 x2 0
1
e
2
9.5 9.5 x1
x1 0
1
16
1
e
16
x2 0
9.5 9.5 x1
0
e
x1 x2
dx2 dx1 / 4
x1 x2
4
dx2 dx1
x1 x2
4
dx2 dx1
0
9.5
9.5
9.5
1
4
9.5e
4e 4 4
4
0
1
9.5
9.5 9.5
e 4 e 4
4
9.5
9.5 9.5
e 4 e 4
Hence, power of the test = 1- 1
4
=
13.5 9.5
e 4 = 0.31 (approx.)
4
Problem 7: Let X have a pdf of the form f ( x) x 1 , 0 x 1, 0 . To test the hypothesis
H 0 : 1 against H1 : 2 , using a random sample x1 , x2 of size 2 and define the critical region as
C { x1 , x2 ;
3
x2 } . Obtain significance level and power of the test
4 x1
Solution:
To test H 0 : 1 against H1 : 2 , given critical region is
3
3
x2 , or x1 x2
4
4 x1
for two independent observations x1 and x2 .
f ( x) x 1 , 0 x 1, 0 , since x1 and x2 are ind. we get,
STATISTICAL INFERENCE
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1
f ( x1 , x2 ) 2 x1 x2
, 0 x1 , x2 1, 0 .
Significance level = P( Rej H 0 / H 0 is true)
= P( x1 x2
3
/ 1)
4
The given critical region is as shaded in the following graph,
3
/ 1)
4
Hence, P( x1 x2
1
1
1
3 3 x x dx dx / 1
2
1 2
1
x1 x2
4
4 x1
1
2
1
3 3
1
1 dx2 dx1
x1 x2
4
4 x1
3
dx1
1
3 1 4 x
x1
4
1
3
3
3 3
x1 log x1 1 log
3
4
4
4 4
4
=
Power of the test
1 3
3
log
4 4
4
= P( Rej H 0 / H1 is true)
= P( x1 x2
STATISTICAL INFERENCE
3
/ 2 )
4
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1
1
1
3 3 x x dx dx / 2
2
1 2
2
1
x1 x2
4
4 x1
1
1
3 3
4 x1 x2 dx2 dx1
x1 x2
4
4 x1
2
3
1
1 4 x1
4 x1 (
)dx1
2
2
3
x1
4
1
4
1
9
3 x ( 2 32 x
1
x1
4
2
1
)dx1
1
x12 9
4 log x1
4 32
3
4
9
3
1 9
power 1 4 log
4
4 64 32
7 9
3
log
16 8
4
4.5. Most Powerful Test:
So far we considered some testing of hypothesis problem with given critical region.
The power and significance level corresponding to a given critical region is calculated.
Now a question arising is, can we find a critical region with maximum power and zero
significance level? But it is not possible. When we are making the probability of type I
error minimum, the probability of type II error increases, thereby power decreases, and
vice versa.
In this dilemma, an approach due to Neymaan and Pearson helps us to achieve a
most powerful test. By their approach, we fix a affordable level of type I error (say 5% or
1% etc) then consider critical regions with the fixed significance level and among them
STATISTICAL INFERENCE
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choose a critical region with maximum power. The test using such a critical region is
known as most powerful test. And the critical region is most powerful critical region (or
Best Critical Region).
Neymaan-Pearson Theorem suggests the method to find a most powerful critical
region to test a simple hypothesis against a simple alternative.
4.6. Neymaan-Pearson Theorem:
Let ( x1 , x2 ,..., xn ) be a random sample from a continuous population with pdf
f ( x, ) . Let S be a subset of the sample space such that, for a constant c,
and for each ( x1 , x2 ,..., xn ) belongs to S,
L0
c
L1
for each ( x1 , x2 ,..., xn ) NOT belongs to S
L0
c
L1
where L0 and L1 are the likelihood of the sample when H 0 is true and H1 is true
respectively. The constant, c is determined so as, P( x1 , x2 ,..., xn S / H 0 ) ,.
Then S is the most powerful critical region with significance level to test the
simple hypothesis H 0 against a simple alternative H1 .
Proof:
Consider H 0 : 0 and H1 : 1 . Let the likelihood of the sample when H 0 is
true is denoted as
L0 f ( x1 , 0 ) f ( x2 , 0 )....... f ( xn , 0 )
n
f ( xi , )
i 1
0
when H 0 is true is denoted as, L1 f ( x1 , 1 ) f ( x2 , 1 )....... f ( xn , 1 )
n
f ( xi , )
i 1
1
Neymaan-Pearson Theorem, for a best critical region S,
L0
c , for each ( x1 , x2 ,..., xn ) S, and
L1
L0
c for each ( x1 , x2 ,..., xn ) S
L1
Consider another critical region S ' , for the given test of size . be the sample space.
STATISTICAL INFERENCE
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P ( x1 , x2 ,..., xn S / H 0 ) L0 dx P ( x1 , x2 ,..., xn S ' / H 0 ) L0 dx
Given
S'
S
But,
S L dx
0
' L dx
L0 dx
S ' S
L dx
and
0
0
S'
S S
L0 dx
S ' S
L0 dx
L0 dx
S ' S
' L dx
0
S S
----- (1)
S ' S
The power of the critical region,
=
S L dx
1
L1dx
'
L0
dx L1dx
c
S S '
S ' S
S S
By (1),
'
S S
L0
dx
c
'
S S
S L dx
1
S L dx
1
1
S S
(
L0
c for x1 , x2 ,..., xn S )
L1
(
L0
c for x1 , x2 ,..., xn S )
L1
L0
dx
c
L0
dx L1dx
c
S S '
'
S S
'
L1dx
S S
STATISTICAL INFERENCE
' L dx
' L dx
1
S S
S L dx
1
' L dx
1
S
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ie.,
power of the critical region S Power of the critical region S ' ,
Hence the
critical region satisfying the conditions of Neymaan-Pearson theorem is the best or most
powerful critical region.
Problem 1: Use Neymaan-Pearson Theorem to find a most powerful test with significance level
for testing the hypothesis H 0 : 0 against, H1 : 1 , ( 1 0 ) using a random sample
1
( x )2
1
e 18
, x .
18
x1 , x2 ,..., xn drawn from the population with pdf f ( x)
Solution:
Given f ( x)
1
( x )2
1
e 18
, x . For the random samples x1 , x2 ,..., xn ,
18
1 n
n
1 18 i1( xi )
the likelihood function L f ( x1 , x2 ,..., xn , )
e
18
n
but ,
But
2
.
n
( xi )2
(x x x )
i 1
i 1
2
i
n
n
i 1
i 1
( xi x ) 2 ( x ) 2 nS 2 n( x ) 2 , where x is the sample
mean and S 2 , the sample variance.
n
1 18 S 2 ( x )2
Therefore, f ( x1 , x2 ,..., xn , ) L
e
18
n
By Neymaan-Pearson theorem, for most powerful critical region S,
L0
c
L1
for ( x1 , x2 ,..., xn ) belongs to S,
n
ie.,
STATISTICAL INFERENCE
1 18 ( S 2 ( x 0 )2 )
e
L0 18
c
L1 1 n n ( S 2 ( x 1 )2 )
18
e
18
n
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e
n 2
( S ( x 0 )2 )
18
n 2
( S ( x 1 )2 )
18
e
e
c
n (( x 0 )2 ( x 1 )2 )
18
c
Taking logarithm on both sides,
n
(( x 0 ) 2 ( x 1 ) 2 ) log c ---- (1)
18
Since 1 0 , dividing both sides of (1) by a negative quantity,
(2 x 0 1 )
``
n
( 1 0 ) , we get,
18
18 log c
n( 1 0 )
18 log c
0 1
for a most powerful critical region, x 1
2 n( )
1
0
18 log c
0 1 c1 ; Then for most powerful critical region,
Let, 1
2 n( )
1
0
x
--------- (2)
c1
The size of the test is considered as . So probability of x1 , x2 ,..., xn to fall in
the critical region, ie., x
c1 , when H 0 is true should be . Using this condition we
can identify the value of the constant c1
c1 / 0 )
ie., P ( x
Since x1 , x2 ,..., xn are random samples from N ( ,3) , the sample mean x ~ N ( ,
x
t
P(
x
3
STATISTICAL INFERENCE
3
n
n
3
)
n
~ N (0,1)
c1
3
n
/ 0 )
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P( t
c1 0
n
3
);
t ~ N (0,1)
----- (3)
From standard normal table a value t can be identified as shown such that,
Comparing (3) and (4),
t
c1 0
3
n
c1 0
3 t
n
Hence the most powerful critical region is,
x 0
3 t
n
Remark: In the derived most powerful critical region, it can be observed that whatever be
the value of in the alternative hypothesis, keeping the condition ( 1 0 ) , the most
powerful critical region is unchanged. That is the most powerful test using this critical
region is uniform.
Hence such a critical region is called Uniformly Most Powerful Critical region or
the test by using such a critical region is Uniformly Most Powerful Test (UMPT).
Problem 2: Use Neymaan-Pearson Theorem to find a most powerful test with significance
level for testing the hypothesis H 0 : 2 0 2 against, H1 : 2 12 ( 12 0 2 ) using a random
1
2
( x )
1
2
e 2
, x ,
sample x1 , x2 ,..., xn drawn from the population with pdf f ( x)
2
where is known.
Solution:
Note that x1 , x2 ,..., xn are random sample taken from N ( , 2 ) . Hence the
STATISTICAL INFERENCE
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1
n
1 2 2
L
e
2
likelihood function,
n
( xi )2
i1
By Neymaan-Pearson theorem, for most powerful critical region S,
L0
c
L1
for ( x1 , x2 ,..., xn ) belongs to S,
ie.,
L0
1n e
L1
0
n
log e
1
n
2 02 i1( xi )
e
n
1
L0 0 2
L1
1
1 2
n
e
2
n
( xi )2
212 i1
1
c
n
1 n
( xi )2
( xi )2
2
2 0 i1
212 i1
1
n
1 n
( xi )2
( xi )2
2
2 0 i1
212 i1
1
c
0n
log ( c n )
1
1
0n
1 n
2
(
x
)
log
(
c
)
i
2
2 0 2 i 1
1n
2 1
Since 12 0 2 ,
1
1 0 2 12
is negative
2
2 0 2
2 0 2 12
2 1
log ( c
n
(x )
i 1
2
i
0n
)
1n
1
1
2
2 0 2
2 1
c1 ( say )
ie., the most powerful critical region is given by,
n
(x )
i 1
STATISTICAL INFERENCE
i
2
c1
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The size of the test is considered as . So probability of x1 , x2 ,..., xn to fall in the critical
region, ie.,
n
(x )
i 1
c1 , when H 0 is true should be .
2
i
Using this condition we can identify the value of the constant c1
n
ie., P (
(x )
i 1
n ( x )2
P i 2
i 1
but we have
2
i
c1 / 2 0 2 )
c1
/ 2 02
2
( xi )
~N (0,1) and
n
P 2( n) / H 0
i 1
( xi ) 2
~ 2( n)
2
i 1
n
c1
02
---- (1)
From the table of 2 ( n ) , the value of 2 :n , as shown below
Comparing (1) and (2),
c1
2 :n
02
Hence for most powerful critical region,
c1 2 :n 0 2
n
(x )
i 1
Thus
the
most
powerful
critical
i
2 :n 0 2
region
H1 : 2 12 ( 12 0 2 ) with significance level is,
STATISTICAL INFERENCE
2
for
testing
H 0 : 2 0 2 against
( xi ) 2
2 :n
2
i 1
n
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EXERCISES
1. Let p be the probability that a coin will fall head in a single toss in order to test
1
3
H 0 : p against H1 : p . The coin is tossed 5 times and H 0 is rejected if more
2
4
than 3 heads obtained. Find the size and power of the test.
2. 10 random samples x1 , x2 ,..., x10 are taken from N ( ,5) . To test H 0 : 0 against
H1 : 2 . The critical region suggested is x1 2 x2 3 x3 ... 10 x10 1 . Obtain the
probability of type-I and type-II errors.
1
3
against H1 : p .
2
4
H 0 is rejected if 60 or more persons are found smokers in a sample of 100 persons.
Compute significance level and power of the test.
3. Let p be the proportion of smokers in a city. To test H 0 : p
4. A single value x is drawn from a normal population N ( m, 5) . The null hypothesis
H 0 : 50 is accepted if x 75 . Otherwise H1 : 60 is considered. Evaluate
significance level and power of the test.
5. A sample of size 10 is taken from a normal population with 1 to test H 0 : 5
against H1 : 6 . The critical region is x 5.52 . Find significance level and power
of the test.
6. Obtain the best critical region for testing H 0 : 0 against H1 : 0 in N ( , )
using a random sample of size n. Also find the power function.
7. In testing H 0 : 0 against H1 : 1 ( 0 ) for the distribution with pdf
f ( x) e
( x )
, 0 x , 0 . Show that the UMP test is of the form
constant and
x
i
x
i
constant.
8. x1 , x2 ,..., xn be a random sample of size n drawn from a population with pdf
f ( x) x 1 , 0 x 1, 0 . Obtain most powerful test for testing H 0 : 0
against H1 : 0 . Examine whether the most powerful test is UMPT.
************************
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CHAPTER 5
LARGE SAMPLE TESTS
5.1. Large Sample Tests:
To test a statistical hypothesis, we construct a test criterion based on an appropriate
test statistic with known probability distribution. If the probability distribution of the
statistic considered is not familiar, it is troublesome to perform the testing procedure.
Fortunately it is known that when the sample size is large, by central limit theorem most
of the statistics are normally or at least approximately normally distributed. That is if U is
a statistic considered, then
t
U E (U )
~ N (0,1) for large n.
SD (U )
U ~ N E (U ), SD(U ) as n become very large, or
This important result may profitably used for the test
construction.
The standard deviation of any statistic is called its standard error. While testing a
hypothesis H 0 : 0 , naturally taking a statistic U with E(U) = . Using standard error of
U, a test statistic t following standard normal distribution can be formed. Then for large
n, the probability that it will fall in any region in its range of variation can be found using
standard normal table.
5.2. Testing mean of a population:
Consider the hypothesis regarding the mean of a population. Let be the
standard deviation of the population.
Consider the hypothesis H 0 : 0 .
The
alternative hypothesis may be (i) H1 : 0 (ii) H1 : 0 or (iii) H1 : 0
Case I: is known:
Let x1 , x2 ,..., xn
be the samples from the population with sample mean x and
sample variance S 2 . In testing of the population mean the sample mean x (which is an
unbiased estimator of population mean) is considered as the best test statistic.
STATISTICAL INFERENCE
If
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H 0 : 0 is to be tested against H1 : 0 , we reject H 0 , if x a constant c. If is the
given significance level, to find the constant c such that P( x c / H 0 ) , that is
P ( x c / 0 ) .
Since the samples are taken from a population with mean and standard
deviation , x ~ N ( ,
) for large n. That is, z
n
x
n
~ N (0,1) for large n.
P ( x c / 0 )
x
P(
P( z
n
c
c 0
n
n
/ 0 )
) , z ~ N (0,1)
---- (1)
From the table of standard normal distribution, one can get a t , such that
P ( z t ) . Hence from (1) ,
c 0
n
c t
t
0
n
Then the test criterion is to reject H 0 : 0 against H1 : 0 , when, x t
ie.,
reject H 0 when t
x 0
n
0
n
t
If H 0 : 0 is to be tested against H1 : 0 , we reject H 0 , if x a constant c. If is the
given significance level, to find the constant c such that P ( x c / H 0 ) , that is
P ( x c / 0 ) .
Therefore,
STATISTICAL INFERENCE
P( z
c 0
n
) , z ~ N (0,1)
----- (2)
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Find a - t , from standard normal table, such that P ( z t ) . Hence from (2) ,
c 0
n
t .
Then the test criterion is to reject H 0 : 0 against H1 : 0 , when, x t
ie.,
reject H 0 when t
x 0
n
0
n
t
If H 0 : 0 is to be tested against H1 : 0 , we reject H 0 , if | x 0 | c a constant c. If
is the significance level, to find the constant c such that P(| x | c / H 0 ) ,
that is P (| x 0 | c) . That is,
P( z
c
n
) , z ~ N (0,1)
---------- (3)
Standard normal table gives value of t , such that
2
c n
t
2
Hence, from (3)
c t
2
n
Then the test criterion is to reject H 0 : 0 against H1 : 0 , when, | x 0 | t
2
ie.,
reject H 0 when
STATISTICAL INFERENCE
t
x 0
n
n
t .
2
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Problem: The average height of a sample of 400 college students is found to be 4.75 ft. The
standard deviation of the population is believed to be 1.5. Does the data contradict the hypothesis
that the mean height of students is 4.48 ft at 1% level of significance?
Solution:
Let be the mean height of the students. To test whether the data consists of 400
samples contradict the belief that average height is 4.48
To test H 0 : 4.48 against H1 : 4.48
Test criterion is t
Here, t
x 0
n
4.75 4.48
.
400
1.5
3.6
The test procedure is to reject H 0 if t t .
2
Given 0.01 , from std normal table, t = 3
2
Then, here
t t . So reject H 0 at 1% level. That is the data contradict the assumption
2
that the average height is 4.48.
Case II: is Unknown:
We have sample variance S 2 as a consistent estimator of 2 . As n becomes large
E ( S 2 ) 2 . Here we consider large sample, hence the value of S 2 is considered as the
value of 2 and can perform the test as already described.
Problem: An insurance agent has claimed that the average age of policy-holders who insure
through him is less than the average for all agents, which is 30.5 years. A random sample of policyholders who had insured through him gave the following age distribution:
Age:
No. of persons:
16 – 20
12
21 – 25
22
26 – 30
31 – 35
36 – 40
20
30
16
Test the claim at 5% level of significance.
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Solution:
Let be the mean age of the policy-holders insured by the agent. A sample of 100
policy-holders insured by the agent is given.
To test H 0 : 30.5 against H1 : 30.5 .
The
population
standard deviation of the age of policy-holders is not given.
Since it is a large sample, sample standard deviation is approximated by the population
standard deviation.
Age
No. of persons Mid-x
fx
fx 2
x
f
16 – 20
12
18
216
3888
21 – 25
22
23
506
11638
26 – 30
20
28
560
15680
31 – 35
30
33
990
32670
36 - 40
16
38
608
23104
2880 86980
x
1
N
fx
i i
i
S .D.
1
N
1
2880 28.8
100
f x x
2
2
i i
i
1
2
86980 28.8
100
869.8 829.44 40.36
6.35
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Since the given sample is of large size, the sample standard deviation is
approximated to the population standard deviation.
Corresponding to 5% level of significance, from standard normal table we get
t 1.65 . The test criterion is to reject H 0 , if, t t , where t
Now
t
28.8 30.5
6.35
100
x 0
n
s
2.68
Here, the calculated value of t = - 2.68 < - t = -1.65. Hence reject H 0 : 30.5 at
5% level of significance. That is the agent’s claim is acceptable.
5.3. Testing the inequalities of means of two populations:
Let 1 , 2 be the means and 1 , 2 be the standard deviations of two populations
and let to test H 0 : 1 2 .
If
H1 : 1 2 , reject H 0 if x1 x2 c , where x1 and x2 are the sample means of n1
samples from first population and n2 samples from the second population taken
independently. Let the significance level is , then to consider c such that
P( x1 x2 c / H 0 )
If t x1 x2 , z
t E (t )
~ N (0,1) for large n. Then,
SD(t )
x1 x2 1 2 c 1 2
P
/ H 0 .
2
2
2
2
1 2
1 2
n
n
n
n
1
2
1
2
This implies that
c
P z
12 2 2
n1
n2
STATISTICAL INFERENCE
, from standard normal table P ( z t )
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c
12 2 2
n1
n2
t ;
so, c t
Hence reject H 0 : 1 2 against H1 : 1 2 , if
The critical region is
t
12 2 2
n1
n2
x1 x2 t
x1 x2
12 2 2
n1
n2
12 2 2
n1
n2
t
In similar way,
Reject H 0 : 1 2 against H1 : 1 2 , if., t t
And,
Reject H 0 : 1 2 against H1 : 1 2 , if,
t
t
2
Remark: If 1 , 2 are unknown, since the sample size is large, the values of sample
standard deviations S1 , S 2 are approximated to perform the testing .
If 1 , 2 are unknown and in addition it is assumed that = 1 = 2 , then the
value of is approximated by
n1S12 n2 S 2 2
.
n1 n2
Problem.1: A sample of 400 men from South India has a mean height of 170 cms. and a standard
deviation of 30 cms. while a sample of 200 men from North India has a mean height of 178 cms
with a standard deviation of 32 cms. Do the data indicate that North Indians are on the average
taller than the South Indians?
Solution:
Large sample of sizies 400 and 200 respectively are taken from two populations and
their mean is found to be 170 and 178. The sample standard deviations are 30 and 32
respectively.
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Let 1 be the mean height of south Indians and 2 be that of North Indians. To
test H 0 : 1 2 against H1 : 1 2 .
The test statistic used is t
x1 x2
12 2 2
n1
n2
. Since the sample sizes are large, then if
12 and 2 2 are unknown, they are approximated b the sample variances.
Hence here, t
170 178
302 322
400 200
= -2.94
Reject H 0 , if t t
Consider a significance level of 5%. Then, from standard normal table t 1.645 .
Therefore here, t= -2.94 < t 1.645 . Hence reject H 0 . That is the data indicates that
North Indians are on the average taller than South Indians.
Problem.2: The mean height of 50 male students who showed above average participation in
college athletics was 68.2 inches with a standard deviation of 2.5 inches; while 50 male students
who showed NO interest in such participation had a mean height of 67.5 inches with a standard
deviation of 2.8 inches.
i.
Test the hypotheses that male students who participate in college athletics are taller
than other male students.
ii.
By how much should the sample size of each of the two groups be increased in
order that the observed difference of 0.7 inches in the mean heights significant
at 5% level of significance.[Assume samples are of equal in size].
Solution:
i.
Let X 1 and X 2 are the variables representing respectively, the heights of the
students who showed and who not showed interest in athletics. Let n1 be the size of
sample with mean x1 and S.D. s1 , taken from the first set. And n2 be that of the second
set with sample mean x2 and S.D s2
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Given, n1 = 50, x1 = 68.2 and s1 = 2.5
; n2 = 50 , x2 = 67.5 and s2 = 2.8.
Let 1 and 2 be the population mean of the sets of students considered. To test
the hypothesis H 0 : 1 2 against H1 : 1 2 .
x1 x2
The test statistic used is t
12 2 2
n1
n2
. Since the sample sizes are large, then if
12 and 2 2 are unknown, they are approximated b the sample variances.
Hence here, t
68.2 67.5
2
2
2.5 2.8
50
50
=
0.7
1.319 .
0.2818
Reject H 0 , if t t
For 5% significance level, from standard normal table, t 1.645 .
But here t t .Hence we accept H 0 . That is, the data is not significant to believe
the first set students are taller than that of the second set.
ii.
Let n be the size of the samples taken from the two sets of students.
The difference of 0.7 units in the mean heights of the samples of sizes n, will
become significant if t t , for that value of n.
At 5% significance level, the data become significant if,
t
68.2 67.5
2.52 2.82
n
n
1.645
0.7
1.645
14.09
n
2
STATISTICAL INFERENCE
1.645
n
14.09 77.81 .
0.7
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That is, when the numbers of samples from two sets are 78 each, the difference 0.7
in the sample mean will become significant. Hence the sample sizes should be increased
by 28, in order that the observed difference of 0.7 inches in the mean heights to become
significant at 5% significance level.
Problem 3: A sample of 200 students from college ‘A’ scored mean mark of 65 with standard
deviation 8 for mathematics in a university examination. Another sample of 100 students from
college ‘B’ scored a mean mark of 60 in the same paper with a standard deviation 6. Does the data
indicate any significance difference between the colleges in terms of the performance in mathematics
paper? Assume the S.D’s are same. (sig. level 5%).
Solution:
Let 1 be the mean marks of the students of college ‘A’ and 2 be that of college
‘B’. To test H 0 : 1 2 against H1 : 1 2 .
Number of sample from first college, n1 200 , x1 65 , s1 8
Number of sample from college ‘B’, n2 100 , x2 60 , s2 6
The S.D’s of the populations are unknown and assumed same. Then the
common standard deviations can be approximated by,
200 64 100 36
= 7.393
200 100
In this case, the test statistic is, t
t
Reject H 0 , if t
n1S12 n2 S 2 2
n1 n2
x1 x2
n1S12 n2 S2 2 1 1
n1 n2 n1 n2
65 60
1
200 64 100 36 1
200 100
200 100
=5.49
t . At 5% level of significance, from standard normal table,
2
t 1.96 . Hence here, it can verify that , t
2
STATISTICAL INFERENCE
t .
2
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Then reject H 0 . That is the two colleges are differing significantly.
Problem 4: Electric bulbs manufactured by X and Y companies gave the following result:
Company
No. of bulbs used
Mean life in hours
S.D.
X
100
1300
82
Y
100
1248
93
Using standard error of the difference between means, state whether there is any significant
difference in the life of the two makes.
Solution:
With usual notations, given n1 100 , x1 1300 , s1 82 ; n2 100 , x2 1248 , s2 93 .
Let 1 and 2 be the average life of bulbs by company X and company Y respectively.
To test, H 0 : 1 2 against H1 : 1 2 .
Standard error of the difference between means =
The test criterion is to reject t
12 2 2
n1
n2
t , where, t
2
x1 x2
12 2 2
n1
n2
.
Since the sample sizes are large use corresponding sample standard deviations
instead of 1 and 2
Hence,
t
1300 1248
2
2
82 93
100 100
52
= 4.19
12.4
For 5% significance level, from standard normal population t 1.96 .
2
Here,
t
t . So reject H 0 . There is significant difference in two makes.
2
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5.4. Testing the population proportion:
Consider the human population of a district.
The proportion of smokers of the
population is to be evaluated. The entire population can be divided in to smokers and
non-smokers. Let X denotes the number of smokers out of N members of the population
then, p
X
is the population proportion regarding smoking habit.
N
Consider a particular characteristic of the population and assume to test
H 0 : p p0 based on a sample of size n taken from the population. Consider the value of
x
, which is the sample proportion regarding the particular characteristic.
n
Then reject H 0 against H1 : p p0 , if
x
E( ) p .
n
x
c.
n
Assume the significance level is . Then for the critical region, it is to find c, such
that P (
x
c / H0 ) .
n
x
x
x
x
x
E( )
n E( n ) c E( n )
n ~ N (0,1) for large n
P
/ H 0 , where, z n
x
x
x
SD( )
SD( )
SD( )
n
n
n
c p0
P z
p0 q0
n
x
x
E ( ) p , SD ( ) pq
n
n
from standard normal table P ( z t ) ,
c p0
p0 q0
n
t
c p0 t
Hence the critical region with size is,
p0 q0
n
x
p0 t
n
p0 q0
n
That is reject H 0 : p p0 against H1 : p p0 , with significance level , if
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x
p0
n
t
p0 q0
n
t .
Similarly, Reject H 0 : p p0 against H1 : p p0 , with significance level , if
t t ,
and
Reject H 0 : p p0 against H1 : p p0 , with significance level , if
t
t .
2
Problem: A random sample of 500 pineapples was taken from a large consignment and 65 were
found to be bad. Test the hypothesis that the percentage of bad apples is 20% at 5% level of
significance. Also obtain a 95% confidence interval for the percentage of bad pineapples in the
consignment.
Solution:
Let p be the proportion of bad pineapples in the consignment.
It is to test H 0 : p 0.20 against H1 : p 0.20
x
p0
The test statistic is t n
p0 q0
n
, reject H 0 , if t
65
0.20
500
Given, n = 500, x = 65. Then t
0.20 0.80
500
t .
2
= -3.91
At 5% level of significance, standard normal table gives, t 1.96 .
2
But, here
t
t . Thus reject H 0 : p 0.20 .
2
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P
We have
t
Hence for 0.05 ,
x
pineapples as t
n
2
t / H 0 . ie., 1 P
2
t
t / H 0
2
we get a confidence interval for the proportion of bad
p0 q0 x
, t
n
n 2
p0 q0
n
.
Since the population proportion of bad pineapples is not known approximate p0
by
x
.
n
So a 95% confidence interval for bad pineapples is ,
65
0.13 0.87
65
0.13 0.87
1.96
,
1.96
500
500
500
500
=
0.1005
, 0.1595
Hence,95% confidence limit for percentage of bad pineapples are 10.05,15.95 .
5.5. Testing the equality of proportion of two populations:
Two test equality of proportion ( H 0 : p1 p2 ) of two populations, consider
independently taken n1 samples from first population and n2 samples from the second
population. Let
x1 x2
,
are the sample proportions regarding the character considered, for
n1 n2
the first and second population respectively.
If
H1 : p1 p2
x x
SD 1 1
n1 n1
,
reject
p1q1 p2 q2
.
n1
n2
STATISTICAL INFERENCE
H0
when
x1 x1
c
n1 n1
where
x x
E 1 1 p1 p2
n1 n1
To find with significance level , P (
and
x1 x1
c / H0 )
n1 n1
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x1 x1
x x
E( 1 1 )
n n1
n1 n1
z 1
~ N (0,1) , then,
x x
SD( 1 1 )
n1 n1
For large n1 and n2 ,
x1 x1
p1 p2
c p1 p2
n n1
P 1
/ H0
p1q1 p2 q2
p1q1 p2 q2
n1
n2
n1
n2
P z
c (0)
pq pq
n1 n2
under H 0 : p1 p2 p ( say )
from standard normal table P ( z t ) ,
c (0)
t
pq pq
n1 n2
Then, reject H 0 , against H1 : p1 p2 , if,
c t
t
pq pq
n1 n2
x1 x1
n1 n1
t
pq pq
n1 n2
If H1 : p1 p2 , reject H 0 , if t t
If H1 : p1 p2 , reject H 0 , if
t
t
2
If p is unknown, estimate its value by p
n1 p '1 n2 p '2
x
x
, where p '1 1 and p '2 2 .
n1 n2
n1
n2
Problem 1: A random sample of 400 men and 600 women were asked whether they should like to
have a flyover near their residence. 200 men and 325 women were in favor of the proposal. Test
the hypothesis that proportions of men and women in favor of the proposal are same against that
they are not, at 5% level of significance.
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Solution:
Let p1 be the proportion of women in favor of the proposal and p2 is that of men.
Then the problem is to test H 0 : p1 p2 against H1 : p1 p2 .
Given a sample of size n1 600 from the women, and n2 400 from the group of
men. From the sample the proportion of women in favor of fly over p1'
te second city is p2 '
200
. Assume p1 p2 p , where
400
p
This implies p
n1 p '1 n2 p '2
,
n1 n2
325 200
0.525
600 400
x1 x1
n1 n1
pq pq
n1 n2
The test statistic used is t
325
and that of
600
325 200
0.542 0.5
600 400
=
= 1.304
0.0322
0.525 0.475 0.525 0.475
600
400
For 0.05 , from std normal table t 1.96 .
2
It can be verified that,
t
t . Then accept H 0 .
2
That is men and women do not differ significantly regarding the proposal for the
fly over.
Problem 2: In a year there are 956 births in a town A, of which 52.5% were males, while when
towns A and B are combined; this proportion in a total of 1,406 births was 0.496. Is there any
significance difference in the proportion of male births in the two towns? (sig. level 1%)
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Solution:
Let p1 be the proportion male birth in the town A, and p2 be the proportion of
male births in town B. Then the problem is to test H 0 : p1 p2 against H1 : p1 p2 .
The number of samples from town A n1 956 . Given the sample proportion of
male birth of town A = 0.525.
The combined sample proportion of male birth out of 1,406 births from town A and
B is given as 0.496.
Hence the number of samples from town B, n2 1, 406 956 450 .
From the given combined proportion, p
n1 p '1 n2 p '2
0.496 , the sample
n1 n2
proportion of the town B can be found as,
956 0.525 450 p '2
0.496
1, 406
p '2
The test statistic used is t
0.496 1, 406 956 0.525
= 0.434.
450
x1 x1
n1 n1
pq pq
n1 n2
0.525 0.434
0.496 0.504 0.496 0.504
956
450
0.091
0.091
3.791
0.25 0.25 0.024
956 450
At 0.01 , from std normal table, t = 3. Therefore here
2
t
t .
2
Hence reject H 0 , that the proportion of male birth are equal for both the town.
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Problem 3: A machine produced 20 defective parts in a batch of 400.
After overhauling, it
produced 10 defectives in a batch of 300. Has the machine improved? [ at 5% level]
Solution:
Let p1 be the proportion of defective production before overhauling [repair] and
p2 is that of after overhauling. The machine is said to be improved if the proportion of
defective production is less. Then the problem is to test H 0 : p1 p2 against H1 : p1 p2 .
Given a sample of size n1 400 and n2 300 . From the sample, the proportion of
defective production before overhauling p1'
20
400
and that of after overhauling is
n1 p '1 n2 p '2
10
. Assume p1 p2 p , where p
,
p2
300
n1 n2
'
This implies p
20 10
0.0429
400 300
The test statistic used is t
x1 x1
n1 n1
pq pq
n1 n2
20 10
0.016666
400 300
=
= 1.0769
0.0154762
0.0429 0.9571 0.0429 0.9571
400
300
For 0.05 , from std normal table t 1.645 .
It can be verified that, t
t . Then accept H 0 .
That is the machine has not at all improved after overhauling.
5.6. Goodness of fit:
The chi-square test is one of the simplest and most commonly used non-parametric
tests of significance by Karl Pearson. It is the most suitable test to compare the obtained
STATISTICAL INFERENCE
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set of ‘observed’ frequencies in a given set with a set of theoretical frequencies within
them.
Consider a set of n possible events, arranged in classes or cells. Let these events
occur with frequencies O1 , O2 ,..., On called observed frequencies. Out of N observations the
expected (theoretical) frequency of each possible event can be evaluated from the
knowledge of the probability distribution suggested for the population. Let they are
denoted by E1 , E2 ,..., En .
Our problem is to verify whether the suggestion regarding the
probability distribution of the population is acceptable. That is to test the compatibility
(or discrepancy) of observed and theoretical frequencies or to determine whether the
deviations, if any, of the observed frequencies from the theoretical frequencies, are small
enough to be regarded as due to fluctuation of random sampling or whether they
indicated that the data could not have possibly come from a population giving rise to
theoretical frequencies.
A measure of discrepancy existing between the observed and expected frequencies
can be found by using the test statistic 2 and is given by
2
n
i 1
Oi Ei
Ei
2
, which follows chi-square distribution with (n-1) degrees of
freedom.
Consider significance level , then find 2
from the table of chi-square
distribution for (n-1) d.f. such that, P ( 2( n 1) 2 ) .
If the calculated value of 2 greater than 2 , it is to conclude that the data could
not have possibly come from a population giving rise to theoretical frequencies.
That is our hypothesis regarding the probability distribution of the population is
rejected. In other words the assumed probability distribution is not fitting in good to the
population considered or the fit is not good.
Conditions for validity of 2 test:
1. The sample size n should be large (say > 50)
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2. The theoretical frequencies of each class should be at least 5, if that of any class is
less than 5,
that class should be combined with the adjacent class and the
corresponding frequency is added together. This process (pooling) should be
repeated till the frequency in all classes are greater than 5.
3. The degree of freedom of 2 is one less than the total number of classes.
If r
parameters are estimated using the observations for the calculation of the
theoretical frequencies, then the degree of freedom of 2 is n-r-1, (n is the total
number of classes after pooling).
Problem 1: When the first proof of 392 pages of a book of 1200 pages were read, the distribution
of printing mistakes were found to be as follows:
No.of mistakes per page (x): 0
1
2
3
4
5
6
No. of pages (f)
72
30
7
5
2
1
:275
Solution:
Let X denote the number of printing mistakes per page. To fit a Poisson distribution
for the given data, first to identify the parameter . For a Poisson random variable X,
E(X) = .
Hence, equate the sample mean to the population mean E(X), to get an
estimate of the parameter .
For the given data, Mean, x
fx
i i
1
N
fx
i i
i
0 275 1 72 .... 6 1 189 ,
N = 392
i
x
189
0.482 .
392
Equating sample mean and population mean, we get 0.482 .
Hence, P ( X x)
STATISTICAL INFERENCE
e 0.482 0.482 x
x!
; x 0,1, 2.....
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x
Observed
P(X = x)
frequency
0
275
1
72
2
30
3
7
4
5
5
2
6
Expected frequency
NxP(X=x) (rounded)
e 0.482 0.4820
= 0.6175
0!
242
e 0.482 0.4821
0.2976
1!
e 0.482 0.4822
0.0717
2!
e 0.482 0.4823
0.0115
3!
117
28
e 0.482 0.4824
0.00138
4!
5
e 0.482 0.4825
0.00013
5!
0
e 0.482 0.4826
0.00001
6!
0
1
0
N = 500
500
Now,
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No.
of Observed
Expected
mistakes(x) frequency Oi
Oi Ei
2
frequency Ei
Oi Ei
Ei
0
275
242
1089
4.5
1
72
117
2025
17.31
2
30
28
4
0.143
144
28.8
3
4
5
7
5
17
3
2
2
5
0
5
0
0
6
392
50.753
From the table,
2
i
Oi Ei
Ei
2
50.753 . We know, here 2 follow chi-
square distribution with (4-1-1) =2 degrees of freedom. (4 classes are considered after
pooling and 1 d.f. is lost due to the estimation of the parameter for calculating theoretical
frequencies)
From chi-square table for 2 d.f., and for significance level 0.05 , 2 5.99 .
Here, 2 2 , that is the data is not matching with the hypothesis considered.
Hence we reject the hypothesis that X follows Poisson distribution.
Problem 2: A survey of 800 families with four children each revealed the following distribution:
Number of Boys:
0
1
2
3
4
Number of Girls:
4
3
2
1
0
178
290
236
64
Number of families:
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Is this result consistent with the hypothesis that male and female births are equally
probable?
Solution:
Here we have to check whether the given data support that male and female births
are equally probable wit probability of male p 0.5 .
Let X denote the number of boys in a family.
Expected number of families with x male children out of 800 families with 4 Childs,
and with probability of male p 0.5 , can be found by 800 P ( X x ) , where P ( X x ) can
be get using binomial probability law as,
P ( X x ) 4C x (0.5) x (0.5) 4 x , x 0,1, 2,3, 4 .
Expected number of families with no male child = 800 P ( X 0)
800 4C0 (0.5) 0 (0.5) 4 0 800 (0.5) 4 50 .
Expected number of families with one male child = 800 P ( X 1)
800 4C1 (0.5)1 (0.5) 4 1 800 4 (0.5) 4 200 .
Expected number of families with two male child = 800 P ( X 2)
800 4C2 (0.5) 2 (0.5) 4 2 800 6 (0.5) 4 300 .
Expected number of families with two male child = 800 P ( X 3)
800 4C3 (0.5)3 (0.5) 4 3 800 4 (0.5) 4 200 .
Expected number of families with two male child = 800 P ( X 4)
800 4C4 (0.5) 4 (0.5) 4 4 800 (0.5) 4 50 .
Now, the expected and observed frequencies of number of boys is,
Number of Boys:
0
1
2
3
4
Observed Number of families ( Oi ):
32
178
290
236
64
Expected Number of families ( Ei ):
50
200
300
200
50
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No. of male Observed
Expected
Oi Ei
2
Oi Ei
frequency Oi
frequency Ei
0
32
50
324
6.48
1
178
200
484
2.42
2
290
300
100
0.33
3
236
200
1296
6.48
4
64
50
196
3.92
800
800
births
From the table,
2
i
Oi Ei
Ei
2
Ei
19.63
2
19.63 .
We know, here 2 follow chi-square
distribution with (5-1) =4 degrees of freedom.
From chi-square table for 4 d.f., and for significance level 0.05 , 2 9.49 .
Here, 2 2 , that is the data is not matching with the hypothesis considered.
Hence we reject the hypothesis that male and female births are equally probable.
Problem 3: A sample analysis of examination results of 200 MBA’s was made. It was found that
46 students had failed, 68 secured a third division, 62 secured a second division and the rest were
placed in first division. Are these figures commensurate with the general examination result which
is in the ratio 4 : 3 : 2 : 1 for various categories respectively?
Solution:
Let us consider the null hypothesis as the data commensurate with the general
examination result. Under this hypothesis our expected number of students in each
category can be calculated as follows:
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Expected number of failed students out of 200 = 200
4
80
4 3 2 1
Expected number of students with third division out of 200 = 200
3
60
4 3 2 1
Expected number of students with second division out of 200 = 200
Expected number of students with first division out of 200 = 200
2
40
4 3 2 1
1
20
4 3 2 1
Calculation fro chi-square value follows:
category
Observed frequency
Expected frequency
Oi Ei
Oi
Ei
Ei
Failed
46
80
14.45
Third divn.
68
60
1.067
Second divn.
62
40
12.1
First divn.
24
20
0.8
200
2
28.417
From the table,
2
i
Oi Ei
Ei
2
28.417 .
At 5% significance level, for (4-1) = 3 d.f. , from chi-square table, 2 7.815 .
Here 2 2 . Hence we reject the hypothesis that the data commensurate
with the general examination result.
5.7. 2 test of independence:
Consider two characteristics A and B. To test whether A and B are independent in
nature. Assume the following table giving frequency corresponds to A and B, which is
divided into different classes,
STATISTICAL INFERENCE
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Under the hypothesis, that A and B are independent,
P (An observation to come in Ai th class) =
f i.
( where f..
f..
P (An observation to come in B j th class) =
f. j
f ij )
i j
f..
then, the probability of an observation to come in Ai th
and
Hence the expected number of observations in Ai th
B j th class
= f..
f i . f. j
f.. f..
and
=
B j th class is
f i . f. j
.
f.. f..
f i . f. j
f..
Under the hypothesis,
2
n
i 1
STATISTICAL INFERENCE
Oi Ei
2
Ei
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l
m
i 1 j 1
fij
2
f i . f. j
f..
,
f i . f. j
which
follows
chi-square
f..
distribution with (l 1)(m 1) degrees of freedom.
If the hypothesis that the
characteristics A and B are independent is acceptable, then, calculated value of 2 ,will be
close to zero.
If 2 > 2 , we reject the hypothesis that the characteristics are
independent. ( 2 is the table value of chi-square distribution for (l 1)(m 1) d.f. such
that, P ( 2(l 1)( m1) 2 ) . ).
For a 2 2 contingency table, where the frequencies are a, b, c and d, as given by,
Problem 1:
B1
B2
A1
a
b
ab
A2
c
d
cd
ac bd
; N=a + b + c +d. Show that, 2
N
( a b c d )( ad bc )2
( a b )(c d )(b d )( a c )
Solution:
2
n
i 1
Oi Ei
Ei
2
.
The expected frequencies of each cell is calculated as,
Expected frequency of (1,1)th cell =
( a b )( a c )
N
Expected frequency of (1,2)th cell =
( a b )(b d )
N
Expected frequency of (2,1)th cell =
(c d )( a c )
N
Expected frequency of (1,1)th cell =
(c d )(b d )
N
2
2
2
( a b )( a c )
( a b )(b d ) (c d )( a c )
(c d )(b d )
a
b
c
d
N
N
N
N
2
( a b )( a c )
( a b )(b d )
(c d )( a c )
(c d )(b d )
N
N
N
N
STATISTICAL INFERENCE
2
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School of Distance Education
2
( a b )( a c )
2
a
1 a( a b c d ) ( a b )( a c )
N
but ,
( a b )( a c )
N
( a b )( a c )
N
1
ad bc
N ( a b )( a c )
2
ad bc ;
1
N ( a b )(b d )
2
Similarly, the other terms are becomes,
ad bc
1
N (c d )( a c )
2
ad bc .
1
N (c d )(b d )
2
and
Hence,
2
2
2
2
ad bc
ad bc
ad bc
1 ad bc
N (a b)(a c) (a b)(b d ) (c d )(a c) (c d )(b d )
2
ad bc
2
1
1
1
1
(a b)(a c ) (a b)(b d ) (c d )(a c ) (c d )(b d )
ad bc
2
(b d ) (a c) (a c) (b d )
(c d )(a c )(b d )
(
a
b
)(
a
c
)(
b
d
)
ad bc
2
N
N
(a b)(a c)(b d ) (c d )(a c )(b d )
N
N
N
(c d ) ( a b )
2
ad bc
(a b)(a c)(b d )(c d )
2
(a b c d )(ad bc) 2
(a b)(c d )(b d )(a c)
Problem 1: Two sample polls of votes for two candidates A and B for a public office are taken, one
from among the residents of rural areas and the other from urban. The results are given in the
adjoining table. Examine whether the nature of the area is related to voting preference in this
election.
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Area
Votes for
A
Total
B
Rural
620
380
1000
Urban
550
450
1000
Total
1170
830
2000
Solution:
Considering ‘the nature of the area and voting preference are independent’ as the
null hypothesis.
Under this hypothesis we get the expected frequency for each cell as
follows.
Expected frequency for the (1,1) cell =
1000 1170
585
2000
Expected frequency for the (1,2) cell =
1000 830
415
2000
Expected frequency for the (2,1) cell =
1170 1000
585
2000
Expected frequency for the (2,2) cell =
1000 830
415
2000
The chi- square value =
i
Oi Ei
2
Ei
620 585
585
2
380 415
415
2
550 585
585
2
450 415
2
415
= 10.0891.
At 5% significance level, for (2-1)(2-1) = 1 d.f. , from chi-square table, 2 3.841 .
STATISTICAL INFERENCE
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Here 2 2 . Hence we reject the hypothesis that the nature of the area and
voting preference are independent. That is the nature of area is related with
the voting preference.
Problem 2: The following contingency table gives the classification of 1000 workers in a factory
according to the disciplinary action taken by the management and their promotional experience:
Disciplinary
Promotional experience
action
Total
Promoted Not promoted
Offenders
30
670
700
Non-offenders
70
230
300
Total
100
900
1000
Test whether promotional experience and disciplinary action taken are associated
or not.
Solution:
Considering
‘promotional
experience
and
disciplinary
action
taken’
are
independent as the null hypothesis, the expected frequency in each cell is calculated as
follows.
Expected frequency for the (1,1) cell =
700 100
70
1000
Expected frequency for the (1,2) cell =
900 700
630
1000
Expected frequency for the (2,1) cell =
100 300
30
1000
Expected frequency for the (2,2) cell =
900 300
270
1000
The chi- square value =
i
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Oi Ei
2
Ei
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30 70
2
70
670 630
630
2
70 30
30
2
230 270
2
270
= 84.66.
At 5% significance level, for (2-1)(2-1) = 1 d.f. , from chi-square table, 2 3.841 .
Here 2 2 . Hence we reject the hypothesis that promotional experience and
disciplinary action taken’ are independent.
That is promotional experience and
disciplinary actions are associated.
Problem 3: A group of 200 boys and 100 girls are selected for an IQ test and they are classified as
given below. Examine whether there is any dependency between the intelligence levels and the
gender
Below average
Average
Above average
Genius
Total
boys
86
60
44
10
200
girls
40
33
25
2
100
126
93
69
12
300
Solution:
Here to test whether the gender (boys and girls) and the IQ levels are independent.
Under the hypothesis the two characteristics given are independent, find the
expected frequency in each cell. Then perform the chi-square test.
If gender and IQ are independent,
expected frequency in (1,1) cell=
200 126
= 84
300
Expected frequency in (1, 2)th cell =
200 93
= 62
300
Expected frequency in (1,3) rd cell =
200 69
= 46
300
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Expected frequency in (1, 4)th cell =
Expected frequency in (1,5)th cell =
200 12
= 8
300
100 126
= 42
300
Expected frequency in (1, 6)th cell =
100 93
= 31
300
Expected frequency in (1, 7)th cell =
100 69
= 23
300
Expected frequency in (1,8)th cell =
Below average
100 12
= 4, Now the expected frequencies
300
Average
Above average
Genius
Total
boys
84
62
46
8
200
girls
42
31
23
4
100
126
93
69
12
300
The calculations for the value of 2 are as follows,
2
n
Oi Ei
Ei
i 1
Gender
IQcategory
Oi
2
Ei
Oi Ei
Oi Ei
2
Oi Ei
2
Ei
Boys
Girls
Below average
86
84
2
4
0.048
Average
60
62
-2
4
0.064
Above average
44
46
-2
4
0.087
Genius
10
8
2
4
0.500
Below average
40
42
-2
4
0.095
Average
33
31
2
4
0.129
25
27
2
25
27
2
0
0
0
300
300
Above average
Genius
total
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The calculated value of 2 = 0.923.
The table value of 2 at 5% significance level for (4-1)(2-1)-1= 2 degrees of freedom
( after the pooling of last two classes in one) is 5.991.
Here 2 < 2 . Hence the hypothesis that the two characteristics are independent
is accepted. That is there is no dependency between gender and IQ level.
Problem 4: A marketing agency gives you the following information about the age groups of the
sample informants and their liking for a particular model of scooter which a company plans to
introduce:
Age group of informants
Below 20
Liked
Total
20 – 39
40 – 59
125
420
60
605
75
220
100
395
200
640
160
1000
Disliked
Total
On the basis of the above data, can it be concluded that the model appeal is independent of
the age group of the informants?
Solution:
Considering ‘the model appeal is independent of the age group’ as the null
hypothesis, the expected frequency in each cell is calculated as follows.
Expected frequency for the (1,1) cell =
605 200
121
1000
Expected frequency for the (1,2) cell =
605 640
387.2
1000
Expected frequency for the (1,3) cell =
605 160
96.8
1000
Expected frequency for the (2,1) cell =
395 200
79
1000
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Expected frequency for the (2,2) cell =
395 640
252.8
1000
Expected frequency for the (2,3) cell =
395 160
63.20
1000
Observed frequency Expected frequency
Oi
Oi Ei
2
Ei
Ei
125
121
0.132
420
387.2
2.77
60
96.8
13.99
75
79
0.202
220
252.8
4.25
100
63.2
21.42
42.764
The calculated value of chi- square =
i
Oi Ei
Ei
2
= 42.764
At 5% significance level, for (3-1)(2-1) = 2 d.f. , from chi-square table, 2 5.991 .
Here 2 2 .
Hence we reject the hypothesis the model appeal is
independent of the age group.
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EXERCISES
1. How do you determine the critical region for testing the mean of a population in
large sample case?
2. What is large sample test? How can the equality of two population proportions be
tested?
3. Explain the principle of a test of goodness of fit.
4. What are the conditions for using chi-square test for testing agreement between
theoretical frequencies and observed frequencies?
5. Derive test statistic and test procedure to test the independence of two attributes in
a 2 2 contingency table.
6. An examination was given to 50 students at college A and to 60 students at college
B. At A, the mean grade was 75 with standard deviation of 9 and at B the mean
grade was 79 with standard deviation 7. Is there significant difference between the
performance of the students at A and those at B at 5%
level of significance?
7. The manufacturer of television tubes knows from past experience that the
average life of a tube is 2,000 hours with a standard deviation of 200 hours. A
sample of 100 tubes has an average life of 1,950 hours. Test at the 0.05 level of
significance, if this sample came from a normal population of mean 2,000 hours.
8. A sample of heights of 6,400 Englishmen has a mean of 67.85 inches and S.D. 2.56
inches, while a sample of heights of 1,600 Australians has a mean of 68.55 inches
and S.D. of 2.52 inches. Do the data indicate that Australians are, on the average,
taller than Englishmen?
9. The mean weekly sale of soap bars in departmental stores was 146.3 bars store.
After an advertising campaign the mean sales in 22 stores for a typical week
increased to 153.7 and showed a standard deviation 17.2. Was the advertising
campaign successful?
10. In a locality 110 persons were randomly selected and asked about their educational
achievement. The results are given as follows:
Sex
below SSLC
SSLC
Above SSLC
Male
15
20
25
Female
25
15
10
Can you conclude in light of this sample, education depends on sex?
**********************
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CHAPTER 6
SMALL SAMPLE TESTS
6.1. Small Sample Tests:
When the number of sample is large, by central limit theorem, almost all test
statistics follows normal distribution. Then for testing the hypothesis, critical region can
be obtained with the help of standard normal table. But when the sample is small, it is to
use test statistics with known probability distributions to perform the testing of
hypothesis.
6.2. Tests based on normally distributed test statistics:
(i) To test mean of a normal population:
Let H 0 : 0 is to be tested, where the mean of a normal population with the
known standard deviation . Let a sample of size n is taken from the normal population.
Then the sample mean x follows N ( ,
x n ~ N (0,1)
) . That is t
n
If H1 : 0 , as illustrated in the last chapter, for a given significance level ,
we reject H 0 , if
x 0
t
n
t , where t is from standard normal table such
that P(t t )
If H1 : 0 , as illustrated in the last chapter, for a given significance level ,
we reject H 0 , if
t
x 0
n
t , where t is from standard normal table such
that P(t t )
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If H1 : 0 , as illustrated in the last chapter, for a given significance level ,
we reject H 0 , if
such that P( t
t
x 0
n
t , where t is from standard normal table
2
2
t )
2
(ii) To test the equality means of two normal population with known standard
deviations
Let samples of sizes n1 and n2 are taken from two normal populations N ( 1 , 1 )
and N ( 2 , 2 ) . To test H 0 : 1 2 , where 12 and 2 2 are known.
The test statistic used is, t
x1 x2
12 2 2
n1
n2
, following N(0,1), under H 0 .
Let is the significance level, then
If H1 : 1 2 ,
Reject H 0 , if t t , where t is from standard normal such
that, P( t t )
If H1 : 1 2 ,
Reject H 0 , if
t t , where t is from standard normal
such that, P ( t t )
If H1 : 1 2 ,
Reject H 0 , if
t t , where t is from standard normal
2
such that, P( t t ) .
2
Problem 1: A sample of size 10 taken from a normal population. The sample mean is recorded as
47. Another sample of size 15 gives its mean as 41. Can the samples be regarded as drawn from
the same population of standard deviation 4 . 0.05 .
Solution:
Two samples of sizes 10 and 15 were drawn from a normal population. The sample
means are 47 and 41 respectively. Here to test whether the samples are from normal
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population with standard deviation 4. That is to test whether the data compatible with
the two population means are equal under common standard deviation 4 .
Hence H 0 : 1 2 against H1 : 1 2
The test statistic used is t
x1 x2
12 2 2
n1
n2
47 41
42 42
10 15
= 3.6742.
For, 0.05 , from std normal table t 1.96 . Hence, here
2
t t . Then reject
2
H 0 . That is, the two samples cannot be regarded as coming from same population.
Problem 2: A sample of size 10 of men and another sample of size 12 of women have mean IQ’s
101 and 98 respectively. Assuming that the IQ’s of men and women are independently
and normally distributed with mean 1 and 2 and S.D’s 4 and 3. Examine whether men are on
the average more intelligent than women at 5% level of significance?
Solution:
The test to perform is H 0 : 1 2 against H1 : 1 2
The test statistic used is t
x1 x2
12 2 2
n1
n2
101 98
42 32
10 12
= 1.96.
For 5% significance level, from standard normal table we get t 1.65 .
Here t t . Hence reject H 0 . That is accepting that men are on the average
more intelligent than women.
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6.3. Tests based on statistics following t - distribution:
(i) To test mean of a normal population (When is unknown):
Consider a normal population N ( , ) . To test H 0 : 0 .
If the population
standard deviation is unknown and since the sample size is small, the test based on
normal distribution is not applicable.
Under the assumptions: (i) the population from the sample am taken is normal
(ii) Standard deviation of the population is unknown
(iii)The sample size is small (say<30)
the statistic ,
t
x1 0
n
s
which follows t-distribution with (n-1)
degrees of freedom is considered as the test statistic.
Let is the significance level, then,
If H1 : 1 2 ,Reject H 0 , if
t t , where t is from table of t – distribution for
(n-1) d.f. such that, P ( t( n 1) t )
If H1 : 1 2 ,
Reject H 0 , if
t t , where t is from table of t –
distribution for (n-1) d.f. such that, P ( t( n 1) t ) .
If H1 : 1 2 ,
Reject H 0 , if
t t , where t is from table of t –
2
distribution for (n-1) d.f. such that, P ( t( n 1) t ) .
2
(ii) To test the equality means of two normal population with known standard
deviations: (when population standard deviations 1 and 2 are unknown)
Let samples of sizes n1 and n2 are taken from two normal populations N ( 1 , 1 )
and N ( 2 , 2 ) , where 1 and 2 are unknown. To test H 0 : 1 2 .
Let the sample standard deviations are S1 and S 2 .
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The test statistic used is, t
x1 x2
n S n2 S2 1 1
n1 n2 2 n1 n2
2
1 1
2
, following t-distribution with (
n1 n2 2 ) degrees of freedom.
Let is the significance level, then
If H1 : 1 2 ,
Reject H 0 , if
t t , where t is from t- table for n1 n2 2
d.f. such that, P ( t( n1 n2 2) t ) .
If H1 : 1 2 ,
Reject H 0 , if
t t , where t is from t- table for n1 n2 2
d.f. such that, P ( t( n1 n2 2) t ) .
If H1 : 1 2 ,
Reject H 0 , if
t t , where t is from t- table for n1 n2 2
2
d.f. such that, P ( t( n1 n2 2) t ) .
2
(iii) To test the equality means based on paired observations:
If to test the equality of mean effect of two different treatments on a population, let
n different pairs of units of the population in such a way that each pair should contain
units of the population which are homogenous in nature is selected. Apply treatments to
each of the pair such that, the first treatment (X) is to the first unit of the pair and the
second treatment (Y) to the second unit.
Collect the data u1 , u2 , ...., un for each pair such that, ui xi yi , where xi is the value
of the effect of treatment (X) on the first unit of the ith pair and yi is the value of the
effect of treatment(Y) on the second unit of the ith pair.
To test H 0 : 1 2 , ie., H 0 : 1 2 0 , where 1 is the mean effect of first treatment
and 2 is te mean effect of second . If the mean effect of two treatments is same, we
expect the value of u as zero.
Assume u follow normal distribution with mean 1 2 and unknown standard
deviation then the test statistic suggested to test H 0 : 1 2 0 , is
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t
u 0
n 1
, which follows t ( n1)
Su
Let is the significance level, then,
If H1 : 1 2 ,Reject H 0 , if
t t , where t is from table of t – distribution for
(n-1) d.f. such that, P ( t( n 1) t )
If H1 : 1 2 ,
Reject H 0 , if
t t , where t is from table of t –
distribution for (n-1) d.f. such that, P ( t( n 1) t ) .
If H1 : 1 2 ,
Reject H 0 , if
t t , where t is from table of t –
2
distribution for (n-1) d.f. such that, P ( t( n 1) t ) .
2
Problem 1: The heights of 10 males are found to be 70,66,59,68,62,63,61,60,59,58 inches. Is it
reasonable to think that the average height is more than 62 inches? Test at 5% level of significance.
Solution:
Let X denotes the height of the males and assume it follow normal distribution.
Now to test the hypothesis regarding the mean of the population. That is to test
H 0 : 62 against H1 : 62 .
Sample mean x of the 10 samples given can be calculated. Population S.D. is
unknown.
Hence the test statistic used is, t
x 0
s
d.f.
Here,
x
= 626
62
63
i
x
i
x
70
66
xi x
7.4
3.4 -3.6
xi x
54.76 11.56 12.96 29.16 0.36 0.16 2.56
2
59
STATISTICAL INFERENCE
68
5.4 -0.6
n 1
61
follows t-distribution with (n-1)
1
626
xi
62.6
n i
10
60
59
0.4 -1.6 -2.6 -3.6
58
-4.6
6.76 12.96 21.16
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S2
1
152.4
2
15.24 S 3.9
xi x
n i
10
The test statistic is, t
x 0
n 1
62.2 62
S
10 1
3.9
0.154 .
Consider 5% level of significance. Then the value of t from table of t-distribution
for 9 degrees of freedom is 1.8331.
Here t 0.154 t . Then we accept H 0 . Hence it is not reasonable to think
that the average height is greater than 62 inches.
Problem 2: The The heights of six randomly chosen sailors are in inches : 63,65,68,69,71, and 72.
Those of 10 randomly chosen soldiers are 61,62,65,66,69,69,70,71,72, and 73. Test whether the
data support the claim that the soldiers are on the average taller than sailors.
Solution:
First group contains n1 12 soldiers and the second group contains n2 15
sailors.
Let 1 denote the average height of soldiers and 2 be that of sailors. Now to test
H 0 : 1 2 against H1 : 1 2 .
The test statistics is t
x1 x2
n S n2 S2 2 1 1
n1 n2 2 n1 n2
2
1 1
To calculate x1 , x2 and S12 , S 2 2 . The necessary steps are as follows,
For the group of soldiers
x
i
408
x1
i
x
63
65
68
69
71
72
xi x
-5
-3
0
1
3
4
xi x
25
9
0
1
9
16
2
STATISTICAL INFERENCE
408
68
6
60
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x x
2
1
60
For the group of sailors,
x
678
70
71
n1S12
i
i
Again,
i
x2
i
x
61
xi x
xi x
2
62
65
66
-6.8 -5.8 -2.8 -1.8
69
69
1.2
1.2
2.2
3.2
678
67.8
10
72 73
4.2 5.2
46.24,33.64,7.84,3.24,1.44,1.44,4.84,10.24,17.64,27.04
n2 S 2 2
x x
i
2
2
153.6
153.6
i
t
68 67.8
60 153.6 1 1
6 10 2 6 10
0.0991
At 5% significance level, from t- table, for
(6+10-2)=14 degrees of freedom,
t 1.76 .
Then it can be observed that , t t . Hence we reject H 0 . That is the soldiers are
on average taller than sailors.
Problem 3: In a certain experiment to compare two types of animal foods A and B, the following
results of increase in weights are observed in anumals. The same sets of eight animals were used in
both the foods.
Animal number
1
2
3
4
5
6
7
8
Total
Increase
Food A xi
49
53
51
52
47
50
52
53
407
In weight
Food B yi
52
55
52
53
50
54
54
53
423
Can we conclude that food B is better than food A?
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Solution:
The above problem is to compare average gain in weight with two foods.
Consider 1 and 2 are the average gain in weight by food A and food B. Let ui xi yi .
If food B is good, u should be negative.
Hence the problem is to test H 0 : 1 2 0
against H1 : 1 2 . That is to test H 0 : u 0 against H 0 : u 0 . Test is performed using
paired t-test.
Test statistic used is t
u 0
n 1
Su
.
animals
1
2
3
4
5
6
7
8
xi
49
53
51
52
47
50
52
53
yi
52
55
52
53
50
54
54
53
ui xi yi
-3
-2
-1
-1
-3
-4
-2
0
-16
9
4
1
1
9
16
4
0
44
ui 2 xi yi
2
u
total
1
16
ui =
2
8
n i
Su
2
1
2
ui u
n i
=
1
(44) (2) 2
8
= 1.23
t
2 0
1.23
8 1
=
- 4.30
At 5% sig. level, from table of t-distribution for 7 degrees of freedom, t 1.895 .
Here calculated value of t t . Hence reject H 0 that is we can conclude food B is
better than food A.
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6.4. Tests based on statistics following 2 - distribution:
To test standard deviation of a normal population:
Let H 0 : 0 , where is the standard deviation of a normal population
against H1 : 0 . Let S be the sample standard deviation of a sample of size n taken
from the normal population. The test statistic suggested is,
2
nS 2
,
02
, which follow chi-square distribution with (n-1) degrees of freedom.
For a significance level , reject H 0 , if,
2 2 ,
where 2 is from 2 table for (n-1) d.f. such that,
P ( 2( n 1) 2 ) .
Problem: It is believed that the weight of one of the product of a company is with variance greater
than 0.16 gms. A Sample of eleven items is taken. Their weights (in gms.) are measured as
follows: 2.5, 2.3, 2.4, 2.3, 2.5, 2.7, 2.5, 2.6, 2.6, 2.7, and 2.5. Test the hypothesis
at 1% level of significance.
Solution:
It is to test H 0 : 0.16 against H1 : 0.16
The test statistic is , 2
nS 2
, following chi-square distribution with )n-1)
02
d.f
Reject H 0 , if 2 2 , with confidence coefficient 1 .
Here,
x
i
i
STATISTICAL INFERENCE
= 27.6
x
1
27.6
xi
2.51
n i
11
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x
2.5
xi x
xi x
2.3
-0.01 -0.21
2
2.4
2.3
2.5
-0.11 -0.21 -0.01
2.7
2.5
2.6
0.19 -0.01 0.09
2.6
2.7
2.5
0.09
0.19
-0.01
0.0001 0.0441 0.0121 0.0441 0.0001 0.0361 0.0001 0.0081 0.0081 0.0361 0.0001
S2
1
0.1891
2
0.0172
xi x
n i
11
2
nS 2
02
=
11 0.0172
= 1.1825
0.16
At 1% significance level, from the table of chi-square distribution for 10 degrees of
freedom, 2 23.2 . Here 2 2 . Hence we accept H 0 .
6.5. Tests based on statistics following F - distribution:
To test the equality of standard deviations of two normal populations:
To test H 0 : 1 2 against H1 : 1 2 , where 1 and 2 are the standard deviations
of two normal populations. Let S1 and S 2 be the standard deviations of the sample of
sizes n1 and n2 is taken from the two populations.
The test statistics suggest is the ratio of the unbiased estimators of population
variances, that is;
If
n1S12
nS2
> 2 2
n1 1 n2 1
,
n1S12
n 1 n n 1 S12 ,
F 1 2 1 2
n2 S 2
n2 n1 1 S 2 2
n2 1
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n1 1, n2 1
F follows F-distribution with
degrees of freedom under the null
hypothesis.
For a significance level ,
Reject H 0 , if F F , where F is from the table of F- distribution for n1 1, n2 1 degrees
2
2
of freedom, such that P ( F n 1, n
1
2 1
F )
2
If
.
2
n2 S 2 2
nS2
< 1 1 ,
n2 1 n1 1
n2 S 2 2
n 1 n n 1 S2 2 , which follows F-distribution with
Then, consider F = 2 2 2 1
n1S1
n1 n2 1 S12
n1 1
n2 1, n1 1
degrees of freedom.
Reject H 0 , against H1 : 1 2 , if F F , where F is from the table of F- distribution for
2
n2 1, n1 1
2
degrees of freedom, such that P ( F n
2 1, n1 1
F )
2
.
2
If H1 : 1 2 , reject H 0 , if F F , where P ( F F ) , and
If H1 : 1 2 , reject H 0 , if F F ' , where P( F F ' ) .
Problem: Two random samples gave the following results:
Sample
size
Sample means
Sample variance
1
10
15
9
2
12
14
9
Test whether the population variances are same at 10% level of significance.
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School of Distance Education
Solution:
Let the two normal populations are N ( 1 , 1 ) and N ( 2 , 2 ) . If the two samples
are from same normal population, 1 2 and 1 2 . Hence it is to test H 0 : 1 2 and
H 0 : 1 2 .
To perform the test on H 0 : 1 2 , we can use t-test with the small samples taken.
And to perform the test on H 0 : 1 2 , we can use F-test.
But to perform the t-test, the basic assumption is 1 2 . Hence first we have to
test H 0 : 1 2 against H1 : 1 2 .
10 9
n1S12
10 , and
=
10 1
n1 1
12 9
n2 S 2 2
9.82
=
n2 1 12 1
n1S12
n 1 n n 1 S12 ,
Hence the test statistic is F 1 2 1 2
n2 S 2
n2 n1 1 S 2 2
n2 1
n1 1, n2 1
follow F-distribution with
d.f.
Here calculated value of F =
10
1.018
9.82
For 10% of significance level, at (10-1,12-1)= (9,11) d.f. From F-table, F 2.90 .
2
Since calculated value is less than table value of F, we accept H 0 : 1 2 .
Now to test, H 0 : 1 2 against H1 : 1 2 , the test statistics is
t
STATISTICAL INFERENCE
x1 x2
n1S12 n2 S2 2 1 1
n1 n2 2 n1 n2
.
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School of Distance Education
Here, t
15 14
10 9 12 9 1 1
10 12 2 10 12
= 0.792.
For 10% significance level, at 20 d.f., t 2.086 . It is observed that | t | t .
2
2
Hence we accept H 0 : 1 2 . Therefore it can be concluded that the two samples are
coming from same normal population.
EXERCISES
1. Describe the method of testing a simple hypothesis H 0 : 0 against the
alternative H1 : 0 in a normal population N ( ,1) when sample size is small.
2. Stating the assumptions, explain t-test for testing mean of a normal population
3. Stating your assumptions, explain t-test to test the equality of means of two
independent normal populations.
4. Explain how t-test is used for paired comparison of difference of means.
5. Explain the uses of t-distribution and F-distribution in testing hypothesis.
6. Discuss the different applications of chi-square as a test statistic.
7. Describe the procedure for testing the equality of variances of two normal
populations.
8. Two different diets ‘A’ and ‘B’ were administered to two different groups of pigs.
The gains in weight by the diets are given below
Diet ‘A’ : 25, 32, 30, 34, 24, 14, 32, 24, 30, 31,
35, 25
Diet ‘B’ : 44, 34, 22, 10, 47, 31, 40, 30, 32, 35, 18, 21, 35 29 22
Test whether the diets differ significantly as far as their effects on increasing weight
is concerned.
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9. Intelligence tests on groups of boys and girls gave the following results:
Group
Mean score
S.D.
Number tested
Boys
70
20
250
Girls
75
15
150
Test whether there is significant difference between the average scores of boys
and girls. Obtain 5% confidence interval for the difference in average scores.
10. Two independent groups of 10 children were tested to find how many digits they
could repeat from memory after hearing them. The results are as follows:
Group A: 8
6
5
7
6
8
7
4
5
6
Group B: 10
6
7
8
6
9
7
6
7
7
Is the difference between the mean scores of the two groups significant?
11. Two random samples of size 8 and 11 drawn from two normal populations are
characterized as follows:
Sample size
8
Sum of observations
9.6
11
Sum of squares of observations
61.52
16.5
73.26
Examine whether the two samples came from populations having same
variance at 5% level of significance.
12. The nicotine content (in mg.) of two samples of tobacco were found to be as
follows:
Sample A: 24
27
26
21
25
Sample B:
30
28
31
22
27
36
Can it be said that the two samples come from the same normal population?
****************
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SYLLABUS
Course-III: Statistical Inference
Module 1: Sampling Distributions; Random sample from a population distribution,
sampling distribution of a statistic, standard error, sampling from a normal population,
sampling distributions of the sample mean and variance, Chi-square, Student’s t and F
distributions- derivations, simple properties and inter relationships.
Module 2: Theory of Estimation: Point estimation, desirable properties of a good
estimator, unbiased, consistency, sufficiency, statement of Fisher-Neyman factorization
criterion, efficiency. Method of estimation, method of moments, method of maximum
likelihood-properties estimators obtained by these methods
Module 3: Interval estimation: Interval estimates of mean, difference of means, variance,
proportions and difference of proportions-Large and small sample cases.
Module 4: Testing of Hypothesis: Concept of testing hypothesis, simple and composite
hypothesis, type I and type II errors, critical region, level of significance and power of a
test. Neymann-Pearson approach-Large sample tests concerning mean, equality of means,
properties, equality of proportions, Small sample tests based on t distribution for mean,
equality of means and paired data. Tests based on F distribution for ratio of variances.
Test based on chi square-distribution for variance, goodness of fit and for independence of
attributes
Books for reference:
1. V.K.Rohatgi : An introduction to probability theory and Mathematical Statistics,
Wiely Eastern
2. S.C.Gupta and V.K.Kapoor : Fundamental of Mathematical Statistics, Sulthan
Chand and Sons
3. Mood A.M, Graybill.F.A. and Boes.D.C : Introduction to Theory of Statistics,
McGrow Hill.
4. John E Freund : Mathematical Statistics : Pearson Education (India), New Delhi
*********
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