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Chapter 7
Statistical Inference: Estimating a
Population Mean
Statistical Inference
Statistical inference is the process of reaching
conclusions about characteristics of an entire
population using data from a subset, or
sample, of that population.
Simple Random Sampling
Simple random sampling is a sampling
method which ensures that every
combination of n members of the
population has an equal chance of being
selected.
Figure 7.1 A Table Of Uniformly
Distributed Random Digits
1
6
8
7
0
5
3
4
9
9
2
9
4
8
7
6
4
3
9
0
5
3
6
4
7
3
6
6
5
9
2
6
8
1
8
0
1
8
1
7
1
8
0
4
1
4
5
9
2
0
6
3
2
5
2
7
0
2
6
1
3
2
4
3
8
3
2
8
5
1
4
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3
3
4
0
2
8
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5
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0
7
1
0
2
6
6
1
0
1
1
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5
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0
5
5
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0
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1
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1
7
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0
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1
0
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7
1
3
1
1
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2
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9
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1
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0
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2
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2
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1
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6
1
8
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9
0
Sample Data
Sample
Member
1
2
3
4
5
.
.
.
49
50
Population
ID
1687
4138
2511
4198
2006
.
.
.
1523
0578
Hrs. of Study
Time(x)
20.0
14.5
15.8
10.5
16.3
.
.
.
12.6
14.0
Figure 7.2
Bar Chart Showing the
Population Study Time
Distribution
P(x)
1/4 = .25
Study Time
x
Figure 7.3 Sampling Distribution for
for Our Small-Scale Illustration
x
P(
x
15
.167
20
.167
25
.333
30
.167
35
.167
)
Figure 7.4 Bar Chart Showing the
Sampling Distribution of x
P( x )
.333
.167
15
20
25
30
35
Sample Mean Study Time (hrs)
x
The Sampling Distribution of the
Sample Mean
The sampling distribution of the sample
mean is the probability distribution of all
possible values of the sample mean, x ,
when a sample of size n is taken from a
given population.
Key Sampling Distribution Properties
•
•
•
For large enough sample sizes, the shape of
the sampling distribution will be approximately
normal.
The sampling distribution is centered on m, the
mean of the population.
The standard deviation of the sampling
distribution can be computed as the population
standard deviation divided by the square root
of the sample size.
Figure 7.5 The Shape of the Sampling
Distribution When Sample Size is
Large (n > 30)
x
Central Limit Theorem
As sample size increases, the sampling
distribution of the sample mean rapidly
approaches the bell shape of a normal
distribution, regardless of the shape of the
parent population.
Figure 7.6
Implications of the
Central Limit Theorem
Population Shapes
x
x
x
The Sampling Distribution of the Sample Mean
n=2
n=5
n = 30
x
x
x
x
x
x
x
x
x
Small Samples
In small sample cases (n<30), the
sampling distribution of the sample mean
will be normal if the shape of the parent
population is normal.
Figure 7.7 The Center of the Sampling
Distribution of the Sample Mean
E(
x)=m
x
Standard Deviation of the
Sampling Distribution of the
Sample Mean
 x

n
(7.1)
Figure 7.8
Standard Deviation of the
Sampling Distribution of the
Sample Mean
 
x
m

n
Figure 7.9
Sampling Distribution of the Sample Mean
for Samples of Size n = 2, n = 8, and n = 20
Selected from the Same Population
Population Distribution
x
Sampling Distribution
n = 20
n=8
n=2
x
Standard Deviation of the Sampling
Distribution of the Sample Mean
(When sample size is a large
fraction of the population size)


x

n
N n
N 1
(7.2)
Interval Estimate of
a Population Mean
x z 
n
(7.3)
Factors Influencing Interval Width
1. Confidence—that is, the likelihood that the interval will
contain m. A higher confidence level will mean a larger z,
which, in turn, will mean a wider interval.
2. Sample size, n. A larger sample size will produce a
tighter interval.
3. Variation in the population, as measured by. The
greater the variation in the population values, the wider
the interval.
Figure 7.10
Intervals Built Around Various
Sample Means from the
Sampling Distribution
x
m
x1
x2
x3
x4
x5
m
Figure 7.11
Standard Error vs.
Margin of Error
x
+
Margin
of Error
z x
Standard
Error
Margin of Error
The margin of error in an interval estimate
of m measures the maximum difference we
would expect between the sample mean
and the population mean at a given level
of confidence.
Figure 7.12 General Comparison of the t
and Normal Distributions
Normal
distribution
t distribution
Interval Estimate of m
When s Replaces 
 s 
x  t

 n
(7.4)
Figure 7.13 Comparison of the t and Normal
Distributions as Degrees of
Freedom Increase
Normal
Distribution
t with 15 degrees of freedom
t with 5 degrees of freedom
Basic Sample Size Calculator
 z 
n 
E 
2
(7.5)
Sample Size when
n/N > .05
n 
n
n
1
N
(7.6)