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ADVANCED BIO-STATISTICS ST.PAULS UNIVERSITY
: Statistical Inferences:
Estimation and Hypotheses Testing:
Introduction:
There are two main purposes of statistics;
• Descriptive Statistics: Organization &
summarization of the data
• Statistical Inference:
Answering research questions about some unknown
population parameters.
⇒ (1) Estimation:
Approximating (estimating) the actual values
of the unknown parameters;
ƒ Point Estimation
ƒ Interval Estimation (or Confidence
Interval: C. I.)
⇒ (2) Hypothesis Testing:
Answering questions about the unknown
parameters of the population (confirming or
denying some conjectures or statements about
the unknown parameters)
We will consider two types of population parameters:
(1) Population mean (for quantitative variables):
µ =The mean of some quantitative variable.
Example:
¾ The mean life span of some bacteria.
¾ The income mean of dentists in Saudi Arabia.
¾ The mean of number of times a Saudi child visit the
pediatrician during the winter season.
(2) Population proportion (for qualitative variables):
π=no. of elements in the population with some specified characteristic
Total no. of elements in the population (population size)
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ADVANCED BIO-STATISTICS ST.PAULS UNIVERSITY
Example:
  The proportion of diabetic patients in Saudi Arabia 
 The proportion of smokers in Riyadh 
 The proportion of females in Saudi Arabia 
5.2 Estimation:
5.2.1 Estimation of Population Mean µ:
We are interested in estimating the mean of a certain population (The
mean of a certain quantitative variable) In this section, we are interested
in estimating the mean of the population (µ . )
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ADVANCED BIO-STATISTICS ST.PAULS UNIVERSITY
(i) Point Estimation of µ:
A point estimate is a single number used to estimate (or
approximate) the true value of .
„
Draw a random sample of size n from the population:
x1, x2 ,K, xn
Compute the sample mean:
(ii) Interval Estimation of µ:
Confidence Interval (C. I. ) of µ:
An interval estimate of
is an interval (L,U) containing
the true value of "with a probability of 1 α ".
1−α is called the confidence coefficient (confidence level) L =
lower limit of the confidence interval
U= upper limit of the confidence interval
„
Draw a random sample of size n from the population
x , x ,K, x
1
2
n
and apply one of the following results.
Result (1): (For the case when is known)
(a) If X 1 , X 2 K, X n is a random sample of size n from a normal
distribution with mean and known variance σ2 , then:
A (1−α 100%) confidence interval for is:
„
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ADVANCED BIO-STATISTICS ST.PAULS UNIVERSITY
(b) If
X 1 , X 2 K, X n is a random sample of size n from any
distribution with mean
and known variance σ2 , and if the
sample size n is large (n ≥30 )
An approximate (1−α 100%) confidence interval for is:
, then:
Note that: We are (1−α 100%) confident that the true value of µ
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ADVANCED BIO-STATISTICS ST.PAULS UNIVERSITY
belongs to the interval
Result (2): (For the case when σ is unknown)
If X 1 , X 2 K, X n is a random sample of size n from a normal
distribution with mean and unknown variance σ2 , then:
A (1−α 100%) confidence interval for is:
„
Where the degrees of freedom is df=ν=n-1.
Note that: We are (1−α 100%) confident that the true value of
belongs to the interval
µ
notice that, in this case we replace σ by S and Z by t.
Notes:
(1) We find
(2) We find
Z
t
1−α2
1−α2
from the Z-table as follows:
from the t-table as follows: (df=ν)
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ADVANCED BIO-STATISTICS ST.PAULS UNIVERSITY
Example:
Suppose that Z ~ N(0,1). Find Z1−α for the following cases:
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(1) α=0.1 (2)
Solution:
(1) For α=0.1:
α
1 − =1 − 0.1
2
2
α=0.05 (3) α=0.01
=0.95
From the table: Z0.95 = 1.645.
(2) For α=0.5:
0. 05
α
1 − 2 =1 − 2
=0.975
From the table: Z0.975 = 1.96.
(3) For α=0.01:
0. 01
α
1 − 2 =1 − 2
=0.995
From the table: Z0.995 = 2.575.
Example:
Suppose that t ~ t(30). Find
Solution: df =
ν= 30
0. 05
α
1 − 2 =1 − 2
=0.975
t
1−α2
=t
t
1−α2
for α=0.05.
=2.0423
0.975
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