Download Continued

Chapters 10 - 11
A Test of the Mean of a Normal Distribution:
Population Variance Unknown
Given a random sample of n observations from a normal population
with mean . Using the sample mean and standard deviation X and
s we can use the following test with significance level ,
(i) To test either null hypothesis
against the alternative
H 0 :   0 or H 0 :   0
H1 :    0
the decision rule is
X-μ0
Reject H 0 if t 
 t n 1,
s/ n
Or equivalently
Reject H 0 if X  X c  0  tn1, s / n
A Test of the Mean of a Normal Distribution:
Population Variance Unknown
(continued)
(ii) To test either null hypothesis
against the alternative
the decision rule is
H 0 :   0 or H 0 :   0
H1 :    0
X-μ0
Reject H 0 if t 
 tn 1,
s/ n
Or equivalently
Reject H 0 if X  X c  0  tn1, s / n
A Test of the Mean of a Normal Distribution:
Population Variance Unknown
(continued)
(iii) To test the null hypothesis
against the alternative
the decision rule is
Reject H 0 if t 
H 0 :   0
H1 :    0
X-μ0
X-μ0
 tn 1, / 2 or Reject H 0 if t 
 t n 1, / 2
s/ n
s/ n
equivalently
Reject H 0 if
X  0  tn1, / 2 s / n or
Reject H 0 if
X  0  tn1, / 2 s / n
where tn-1,/2 is the student t-value for n – 1 degrees of freedom* and upper
tail probability** /2.
The p-values for these tests are computed in the same way as we did for
tests with known variance except that the student t value is substituted***
for the normal Z value. *frígráður **líkur í efri hala ***sett í stað
Chapter 9.4 (327)
Tests of the Population Proportion
(Large Sample Size)
We begin by assuming a random sample of n observations from a
population that has a proportion  whose members possess a
particular attribute (tiltekið viðhorf) . If (1 - ) > 9 and the sample
proportion is p the following tests have significance level :
(i) To test either null hypothesis
H 0 :    0 or H 0 :    0
against the alternative (valtilgátunni)
H1 :    0
the decision rule is (ákvörðunarreglunni)
Reject H 0 if Z 
p 0
 Z
 0 (1   0 ) / n
Tests of the Population Proportion
(Large Sample Size)
(Continued)
(ii) To test either null hypothesis
against the alternative
the decision rule is
H 0 :    0 or H 0 :    0
H1 :    0
p 0
Reject H 0 if Z 
  Z
 0 (1   0 ) / n
Tests of the Population Proportion
(Large Sample Size)
(Continued)
(iii) To test the null hypothesis
H0 :    0
against the two-sided alternative
the decision rule is
Reject H 0 if Z 
H1 :    0
p 0
 Z / 2
 0 (1   0 ) / n
or Reject H 0 if Z 
p 0
  Z / 2
 0 (1   0 ) / n
For all of these tests the p-value is the smallest significance level at which the
null hypothesis can be rejected.
Chapter 9.5 (330)
Tests of Variance of a Normal Population
Given a random sample of n observations from a normally
distributed population with variance 2. If we observe the sample
variance sx2, then the following tests have significance level :
(i) To test either the null hypothesis
H 0 :  2   02 or H 0 :  2   02
against the alternative
the decision rule is
H1 :  2   02
Reject H 0 if
(n  1) s x2
 02
  n21,
Tests of Variance of a Normal Population
(continued)
(ii) To test either null hypothesis
H 0 :  2   02 or H 0 :  2   02
against the alternative
the decision rule is
H1 :  2   02
Reject H 0 if
(n  1) s x2
 02
  n21,1
Tests of Variance of a Normal Population
(continued)
(iii) To test the null hypothesis
H 0 :  2   02
against the alternative
H1 :  2   02
the decision rule is
Reject H 0 if
(n  1) s x2
 02

2
n 1, / 2
or
(n  1) s x2
 02
  n21,1 / 2
Where 2n-1 is a chi-square random variable and P(2n-1 > 2n-1,) = .
The p-value for these tests is the smallest significance level at which the
null hypothesis can be rejected given the sample variance.
Some Probabilities for the Chi-Square
Distribution
(Figure 9.5) Section 9.5 page 331
f(2v)
/2
1-
0
2v,1-/2
/2
2v,/2
Chapter 9.6 (334)
Tests of the Difference Between
Population Means: Matched Pairs
Suppose that we have a random sample of n matched (samstæð)
pairs of observations (mælinga/athugana) from distributions with
means X and Y . Let D and sd denote the observed sample mean
and standard deviation for the n differences Di = (xi – yi) . If the
population distribution of the differences is a normal distribution,
then the following tests have significance level .
(i) To test either null hypothesis
H 0 :  x   y  D0 or H 0 :  x   y  D0
against the alternative
the decision rule is
H1 :  x   y  D0
Reject H 0 if
D -D0
 tn 1,
sD / n
Tests of the Difference Between
Population Means: Matched Pairs
(continued)
(ii) To test either null hypothesis
H 0 :  x   y  D0 or H 0 :  x   y  D0
against the alternative
H1 :  x   y  D0
the decision rule is
Reject H 0 if
D -D0
 tn1,
sD / n
Tests of the Difference Between
Population Means: Matched Pairs
(continued)
(iii) To test the null hypothesis
H 0 :  x   y  D0
against the two-sided alternative
the decision rule is
Reject H 0 if
H1 :  x   y  D0
D -D0
 tn1, / 2
sD / n
or
D -D0
 tn1, / 2
sD / n
Here tn-1, is the number for which P(tn-1 > tn-1, ) = 
where the random variable tn-1 follows a Student’s t distribution with (n –
1) degrees of freedom. When we want to test the null hypothesis that the
two population means are equal, we set D0 = 0 in the formulas. P-values
for all of these tests are interpreted as the smallest significance level at
which the null hypothesis can be rejected (hægt er að hafna) given the test
statistic.
Tests of the Difference Between Population
Means: Independent Samples (Known
Variances)
Suppose that we have two independent random samples of nx and ny
observations from normal distributions with means X and Y and
variances 2x and 2y . If the observed sample means are X and Y, then
the following tests have significance level .
(i) To test either null hypothesis
H 0 :  x   y  D0 or H 0 :  x   y  D0
against the alternative
the decision rule is
H1 :  x   y  D0
Reject H 0 if
X  Y -D0
 x2
nx


2
y
ny
 Z
Tests of the Difference Between Population
Means: Independent Samples (Known Variances)
(continued)
(ii) To test either null hypothesis
H 0 :  x   y  D0 or H 0 :  x   y  D0
against the alternative
the decision rule is
H1 :  x   y  D0
Reject H 0 if
X  Y -D0

2
x
nx


2
y
ny
  Z
Tests of the Difference Between Population
Means: Independent Samples (Known Variances)
(continued)
(iii) To test the null hypothesis
H 0 :  x   y  D0
against the alternative
H1 :  x   y  D0
the decision rule is
Reject H 0 if
X  Y -D0

2
x
nx


2
y
ny
  Z / 2
or
X  Y -D0

2
x
nx


2
y
 Z / 2
ny
If the sample sizes are large (n > 100) then a good approximation at
significance level  can be made if the population variances are
replaced by the sample variances. In addition the central limit leads to
good approximations even if the populations are not normally
distributed. P-values for all these tests are interpreted as the smallest
significance level at which the null hypothesis can be rejected given
the test statistic.
Tests of the Difference Between Population
Means: Population Variances Unknown and Equal
These tests assume that we have two independent random samples of
nx and ny observations from normally distributed populations with
means X and Y and a common variance. The sample variances sx2
and sy2 are used to compute a pooled variance estimator
s 2p 
(nx  1) s x2  (n y  1) s y2
(nx  n y  2)
Then using the observed sample means are X and Y, the following tests
have significance level :
(i) To test either null hypothesis
H 0 :  x   y  D0 or H 0 :  x   y  D0
against the alternative
the decision rule is
H1 :  x   y  D0
Reject H 0 if
X  Y -D0
s
2
p
nx

s
2
p
ny
 t nx  n y  2,
Tests of the Difference Between Population
Means: Population Variances Unknown and Equal
(continued)
(ii) To test either null hypothesis
H 0 :  x   y  D0 or H 0 :  x   y  D0
against the alternative
the decision rule is
H1 :  x   y  D0
Reject H 0 if
X  Y -D0
s
2
p
nx

s
2
p
ny
 t nx  n y  2,
Tests of the Difference Between Population
Means: Population Variances Unknown and Equal
(continued)
(iii) To test the null hypothesis
H 0 :  x   y  D0
against the alternative
H1 :  x   y  D0
the decision rule is
Reject H 0 if
X  Y -D0
s
2
p
nx

s
2
p
ny
 t nx  n y  2, / 2
or
X  Y -D0
s
2
p
nx

s
2
p
 t nx  n y  2, / 2
ny
Here tnx+ny-2, is the number for which P(tnx+ny-2, > tnx+ny-2, ) = .
P-values for all these tests are interpreted as the smallest significance
level at which the null hypothesis can be rejected given the test
statistic.
Tests of the Difference Between Population Means:
Population Variances Unknown and Not Equal
These tests assume that we have two independent random samples of nx
and ny observations from normal populations with means X and Y and a
common variance. The sample variances sx2 and sy2 are used. The degrees
of freedom, v, for the student t statistic is given by
2
2
 s x2

sy
(
)

(
)

n y 
 nx
v 2
s y2 2
sx 2
( ) /( nx  1)  ( ) /( n y  1)
nx
ny
Then using the observed sample means are X and Y, the following tests
have significance level :
(i) To test either null hypothesis
H 0 :  x   y  D0 or H 0 :  x   y  D0
against the alternative
the decision rule is
H1 :  x   y  D0
Reject H 0 if
X  Y -D0
2
x
2
y
s
s

nx n y
 tv ,
Tests of the Difference Between Population Means:
Population Variances Unknown and Not Equal
(continued)
(ii) To test either null hypothesis
H 0 :  x   y  D0 or H 0 :  x   y  D0
against the alternative
the decision rule is
H1 :  x   y  D0
Reject H 0 if
X  Y -D0
2
y
s x2 s

nx n y
 tv ,
Tests of the Difference Between Population Means:
Population Variances Unknown and Not Equal
(continued)
(iii) To test the null hypothesis
against the alternative
H1 :  x   y  D0
the decision rule is
Reject H 0 if
H 0 :  x   y  D0
X  Y -D0
2
y
s x2 s

nx n y
 tv , / 2
or
X  Y -D0
2
y
s x2 s

nx n y
 tv , / 2
Here tnx+ny-2, is the number for which P(tnx+ny-2, > tnx+ny-2, ) = .
P-values for all these tests are interpreted as the smallest significance
level at which the null hypothesis can be rejected given the test
statistic.
Chapter 9.7 (346)
Testing the Equality of Population Proportions
(Large Samples)
Given independent random samples of nx and ny with proportion successes px
and py. When we assume that the population proportions are equal, an
estimate of the common proportion is
p0 
nx p x  n y p y
nx  n y
For large sample sizes - - n(1 - ) > 9 - - the following tests have significance
level :
(i) To test either null hypothesis
H 0 :  x   y  0 or H 0 :  x   y  0
against the alternative
the decision rule is
Reject H 0 if
H1 :  x   y  0
( px  p y )
p0 (1  p0 ) p0 (1  p0 )

n
n
 Z
Testing the Equality of Population Proportions
- Large Samples (continued)
(ii) To test either null hypothesis
H 0 :  x   y  0 or H 0 :  x   y  0
against the alternative
the decision rule is
Reject H 0 if
H1 :  x   y  0
( px  p y )
p0 (1  p0 ) p0 (1  p0 )

nx
ny
  Z
Testing the Equality of Population Proportions
- Large Samples (continued)
(iii) To test the null hypothesis
against the alternative
H1 :  x   y  0
the decision rule is
Reject H 0 if
H0 :  x   y  0
( px  p y )
p0 (1  p0 ) p0 (1  p0 )

nx
ny
  Z / 2
or
( px  p y )
p0 (1  p0 ) p0 (1  p0 )

nx
ny
 Z / 2
It is also possible to compute and interpret the p-values for these tests
by calculating the minimum significance level at which the null
hypothesis can be rejected.
Chapter 9.8 (350)
The F Distribution
Given that we have two independent random samples of nx and ny
observations from two normal populations with variances 2x and 2y .
If the sample variances are sx2 and sy2 then the random variable
s x2 /  x2
F 2 2
sy /  y
Has an F distribution with numerator degrees of freedom (nx – 1) and
denominator degrees of freedom (ny – 1). An F distribution with
numerator degrees of freedom v1 and denominator degrees of freedom
v2 will be denoted Fv1, v2 . We denote Fv1, v2,  the number for which
P( Fv1 ,v2  Fv1 ,v2 , )  
We need to emphasize that this test is quite sensitive to the assumption
of normality.
Tests for Equality of Variances from Two
Normal Populations
Let sx2 and sy2 be observed sample variances from independent
random samples of size nx and ny from normally distributed
populations with variances 2x and 2y . Use s2x to denote the larger
variance. Then the following tests have significance level :
(i) To test either null hypothesis
H 0 :  x2   y2 or H 0 :  x2   y2
against the alternative
the decision rule is
H1 :  x2   y2
s x2
Reject H 0 if F  2  Fnx 1,n y 1,
sy
Tests for Equality of Variances from Two
Normal Populations
(continued)
(ii) To test the null hypothesis
H 0 :  x2   y2
against the alternative
H1 :  x2   y2
the decision rule is
s x2
Reject H 0 if F  2  Fnx 1,n y 1, / 2
sy
Where s2x is the larger of the two sample variances. Since either sample
variance could be larger this rule is actually based on a two-tailed test
and hence we use /2 as the upper tail probability. Here Fnx-1,ny-1 is the
number for which
P( Fnx 1,ny 1  Fnx 1,n y 1, )  
Where Fnx-1,ny-1 has an F distribution with (nx – 1) numerator degrees of
freedom and (ny – 1) denominator degrees of freedom.
Chapter 9.9 (354)
Determining the Probability of a Type II Error
Consider the test
against the alternative
Using a decision rule
H 0 :   0
H1 :    0
X-μ0
Reject H 0 if
 Z / 2
σ/ n
or
X  μ0  Z σ/ n  X c
Using the decision rule determine the values of the sample mean that result in
accepting the null hypothesis. Now for any value of the population mean
defined by the alternative hypothesis H1 find the probability that the sample
mean will be in the acceptance region for the null hypothesis. This is the
probability of a Type II error. Thus we consider  = * such that * > 0. Then
for * the probability of a Type II error is
*
X


  P( X  X c |    * )  P[ Z  c
]
/ n
and
Power = 1 - 
Power Function for Test H0:  = 5 against
H1:  > 5 ( = 0.05,  =0.1, n = 16)
Power (1 - )
(Figure 9.13)
1
.5
.05
0
5.00
5.05
5.10

Key Words
 Alternative Hypothesis
 Determining the
Probability of Type II
Error
 Equality of Population
Proportions
 F Distribution
 Hypothesis Testing
Methodology
 Interpretation of the
Probability value or pvalue
 Null Hypothesis
 Power Function
 States of Nature and
Decisions on Null
Hypothesis
 Test of Mean of a Normal
Distribution (Variance
Known)
 Composite Null and
Alternative
 Composite or Simple Null
and Alternative
 Hypothesis
Key Words
(continued)
 Testing the Equality of
Two Population
Proportions (Large
Samples)
 Tests for Difference
Between Population
Means: Independent
Samples
 Tests for Equality of
Variances from Two
Normal Populations
 Tests for the Difference
Between Sample Means:
Population Variances
Unknown and Equal
 Tests for Differences
Between Population
Means: Matched Pairs
 Tests of the Mean of a
Normal Distribution:
Population Variance
Unknown
 Tests of the Population
Proportion (Large
Sample Sizes)
 Tests of Variance of a
Normal Population
 Type I Error
 Type II Error