統計學
授課教師：統計系余清祥
日期：2014年11月27日
第八章：區間估計
Fall 2014
STATISTICS in PRACTICE
• Food Lion is one of the largest
supermarket chains in the USA.
• Food Lion establishes a LIFO
(last-in, first-out) index for each
of seven inventory pools.
• Using a 95% confidence level, Food Lion
computed a margin of error of .006 for the
sample estimate of LIFO index.
• The interval from 1.009 to 1.021 provided a 95%
confidence interval estimate of the population
LIFO index.
• This level of precision was judged to be very
good.
2
Chapter 8 Interval Estimation
• 8.1 Population Mean: σ Known
• 8.2 Population Mean: σ Unknown
• 8.3 Determining the Sample Size
• 8.4 Population Proportion
3
8.1 Population Mean: σ Known -1
• Margin of Error and the Interval Estimate
• Practical Advice
4
Population Mean: σ Known -2
• Example: Lloyd’s Department Store
– Each week Lloyd’s Department Store selects a
simple random sample of 100 customers in order
to learn about the amount spent per shopping trip.
With x representing the amount spent per
shopping trip, the sample mean x provides a
point estimate of µ, the mean amount spent per
shopping trip for the population of all Lloyd’s
customers. Lloyd’s has been using the weekly
survey for several years. Based on the historical
data, Lloyd’s now assume a know value of σ =
$20 for the population standard deviation.
5
Margin of Error and the Interval Estimate
-1
A point estimator cannot be expected to provide the
exact value of the population parameter.
An interval estimate can be computed by adding and
subtracting a margin of error to the point estimate.
Point Estimate +/− Margin of Error
The purpose of an interval estimate is to provide
information about how close the point estimate is to
the value of the parameter.
6
Margin of Error and the Interval Estimate
-2
The general form of an interval estimate of a
population mean is
x ± Margin of Error
7
Interval Estimate of a Population Mean:
σ Known -1
• In order to develop an interval estimate of a
population mean, the margin of error must
be computed using either:
– the population standard deviation σ , or
– the sample standard deviation σ
• σ is rarely known exactly, but often a good
estimate can be obtained based on historical
data or other information.
• We refer to such cases as the σ known case.
8
Interval Estimate of a Population Mean:
σ Known -2
• There is a 1 – α probability that the value of a
sample mean will provide a margin of zα /2σ x
error of or less.
Sampling
distribution
of x
α/2
1 - α of all
x values
zα /2 σ x
µ
α/2
x
zα /2 σ x
9
Interval Estimate of a Population Mean:
σ Known -3
Sampling
distribution
of x
α/2
interval
does not
include µ
1 - α of all
x values
zα /2σ x
µ
x
zα /2σ x
x -------------------------]
interval
includes µ
x -------------------------]
[------------------------- x -------------------------]
[------------------------[-------------------------
α/2
10
Interval Estimate of a Population Mean:
σ Known -4
• Example: Lloyd’s Department Store
– The sampling distribution of x is normally
distributed with a standard error of σ x = 2.
– Because ±1.96σ x = 21.96(2)= 3.92, we can conclude
that 95% of all x values obtained using a sample
size of n = 100 will be within ± 3.92 of the
population mean μ.
11
Interval Estimate of a Population Mean:
σ Known -5
• Example: Lloyd’s Department Store,
Sampling Distribution of Showing the Location of x
Sample Means that are Within 3.92 of μ
12
Interval Estimate of a Population Mean:
σ Known -6
• Example: Lloyd’s Department Store,
Intervals Formed From Selected Sample Means at
Location x1, x2, and x3
13
Meaning of Confidence (σ Known) -1
• Example: Lloyd’s Department Store
– Any sample mean that is within the darkly
shaded region of Figure 8.3 will provide an
interval that contains the population mean μ.
Because 95% of all possible sample means are in
the darkly shaded region, 95% of all intervals
formed by subtracting 3.92 from and adding 3.92
to will include the population mean μ.
14
Meaning of Confidence (σ Known) -2
• Example: Lloyd’s Department Store
– During the most recent week, the quality
assurance team at Lloyd’s surveyed 100
customers and obtained a sample mean amount
spent of x = 82. Using x ± 3.92 to construct the
interval estimate, we obtain 82 ± 3.92. Thus, the
specific interval estimate of μ based on the data
from the most recent week is 82 – 3.92 = 78.08 to
82 + 3.92 = 85.92.
15
Meaning of Confidence (σ Known) -3
• Example: Lloyd’s Department Store
– Because 95% of all the intervals constructed using
x ± 3.92 will contain the population mean, we say
that we are 95% confident that the interval 78.08
to 85.92 includes the population mean μ. We say
that this interval has been established at the 95%
confidence level.
– The value 0.95 is referred to as the confidence
coefficient, and the interval 78.08 to 85.92 is called
the 95% confidence interval.
16
Meaning of Confidence (σ Known) -4
• Interval Estimate of µ
x ± zα /2
where:
σ
n
x is the sample mean
1 – α is the confidence coefficient
zα/2 is the z value providing an area of
α/2 in the upper tail of the standard
normal probability distribution
σ is the population standard deviation
n is the sample size
17
Meaning of Confidence (σ Known) -5
• Values of zα/2 for the Most Commonly Used
Confidence Levels
18
Meaning of Confidence (σ Known) -6
Because 90% of all the intervals constructed using
x ± 1.645σ x will contain the population mean,
we say we are 90% confident that the interval
x ± 1.645σ x includes the population mean µ.
We say that this interval has been established at the
90% confidence level.
The value 0.90 is referred to as the confidence
coefficient.
19
Meaning of Confidence (σ Known) -7
• Example: Lloyd’s Department Store
– For a 95% confidence interval, the confidence
coefficient is (1 – α) = 0.95 and thus, α = 0.05.
– Using the standard normal probability table, an
area of α/2 = 0.05/2 = 0.025 in the upper tail
provides z0.025 = 1.96. With the Lloyd’s sample
mean x = 82, σ = 20, and a sample size n = 100,
20
82 ± 1.96
100
82 ± 3.92
– Thus, the margin of error is 3.92 and the 95%
confidence interval is 82 – 3.92 = 78.08 to 82 + 3.92
85.92.
20
Meaning of Confidence (σ Known) -8
• Example: Lloyd’s Department Store
– The 90% confidence interval for the Lloyd’s
example is
20
82 ± 1.645
100
82 ± 3.29
– At 90%, the margin of error is 3.29 and the
confidence interval is 82 – 3.28 = 78.71 to 82 + 3.28
= 85.29.
21
Meaning of Confidence (σ Known) -9
• Example: Lloyd’s Department Store
– The 99% confidence interval is
82 ± 2.576
82 ± 5.15
20
100
– Thus, at 99% confidence, the margin of error is
5.15 and the confidence interval is 82 – 5.15 =
76.85 to 82 + 5.15 = 87.15.
22
Interval Estimate of a Population Mean:
σ Known -7
• Example: Discount Sounds
– Discount Sounds has 260 retail outlets throughout
the United States. The firm is evaluating a
potential location for a new outlet, based in part,
on the mean annual income of the individuals in
the marketing area of the new location.
– A sample of size n = 36 was taken; the sample
mean income is $41,100. The population is not
believed to be highly skewed. The population
standard deviation is estimated to be $4,500, and
the confidence coefficient to be used in the
interval estimate is 0.95.
23
Interval Estimate of a Population Mean:
σ Known -8
• Example: Discount Sounds
– 95% of the sample means that can be observed are
within ±1.96σ x of the population mean µ.
– The margin of error is:
σ
 4, 500 
=
zα /2
1.96
=


n
 36 
1, 470
Thus, at 95% confidence,
the margin of error is $1,470.
24
Interval Estimate of a Population Mean:
σ Known -9
• Example: Discount Sounds
– Interval estimate of µ is:
$41,100 ± $1,470
or
$39,630 to $42,570
– We are 95% confident that the interval contains
the population mean.
25
Interval Estimate of a Population Mean:
σ Known -10
• Example: Discount Sounds
Confidence
Level
90%
95%
99%
Margin
of Error
Interval Estimate
3.29
3.92
5.15
78.71 to 85.29
78.08 to 85.92
76.85 to 87.15
– In order to have a higher degree of confidence,
the margin of error and thus the width of the
confidence interval must be larger.
26
Interval Estimate of a Population Mean:
σ Known -11
• Adequate Sample Size
In most applications, a sample size of n = 30 is
adequate.
If the population distribution is highly skewed or
contains outliers, a sample size of 50 or more is
recommended.
27
Interval Estimate of a Population Mean:
σ Known -12
• Adequate Sample Size (continued)
If the population is not normally distributed but is
roughly symmetric, a sample size as small as 15
will suffice.
If the population is believed to be at least
approximately normal, a sample size of less than 15
can be used.
28
8.2 Population Mean: σ Unknown
• Margin of Error and the Interval Estimate
• Practical Advice
• Using a Small Sample
• Summary of Interval Estimation Procedures
29
Interval Estimate of a Population Mean:
σ Unknown
• If an estimate of the population standard
deviation σ cannot be developed prior to
sampling, we use the sample standard
deviation s to estimate σ.
• This is the σ unknown case.
• In this case, the interval estimate for µ is
based on the t distribution.
• We’ll assume for now that the population is
normally distributed.
30
t Distribution -1
William Gosset, writing under the name “Student”,
is the founder of the t distribution.
Gosset was an Oxford graduate in mathematics
and worked for the Guinness Brewery in Dublin.
He developed the t distribution while working on
small-scale materials and temperature experiments.
31
Student - William Sealy Gosset (1876 - 1937)
This birth-and-death process is suffering from
labor pains; it will be the death of me yet.
(Student Sayings)
32
t Distribution -2
The t distribution is a family of similar probability
distributions.
A specific t distribution depends on a parameter
known as the degrees of freedom.
Degrees of freedom refer to the number of
independent pieces of information that go into the
computation of s.
33
t Distribution -3
A t distribution with more degrees of freedom has
less dispersion.
As the degrees of freedom increases, the difference
between the t distribution and the standard
normal probability distribution becomes smaller
and smaller.
34
t Distribution -4
t distribution
(20 degrees
of freedom)
Standard
normal
distribution
t distribution
(10 degrees
of freedom)
z, t
0
35
t Distribution -5
• Selected Values From the t Distribution Table
– Entries in the table give t values for an area or
probability in the upper tail of the t distribution.
36
t Distribution -6
(Table )
• For example, with 10 degrees of freedom and a 0.05
area in the upper tail, t0.05 = 1.812.
37
t Distribution -7
For more than 100 degrees of freedom, the standard
normal z value provides a good approximation to
the t value.
The standard normal z values can be found in the
infinite degrees ( ) row of the t distribution table.
38
t Distribution -8
Standard normal
z values
39
Margin of Error and the Interval Estimate:
σ Unknown -1
• Interval Estimate
x ± tα /2
s
n
where:
1 - α = the confidence coefficient
tα/2 = the t value providing an area of α/2 in the
upper tail of a t distribution with n – 1
degrees of freedom
σ = the sample standard deviation
40
Margin of Error and the Interval Estimate:
σ Unknown -2
• Example: Apartment Rents
– A reporter for a student newspaper is writing an
article on the cost of off-campus housing. A
sample of 16 one-bedroom apartments within a
half-mile of campus resulted in a sample mean of
$750 per month and a sample standard deviation
of $55.
– Let us provide a 95% confidence interval estimate
of the mean rent per month for the population of
one-bedroom efficiency apartments within a halfmile of campus. We will assume this population
to be normally distributed.
41
Margin of Error and the Interval Estimate:
σ Unknown -3
• At 95% confidence, α = 0.05, and α/2 = 0.025. t0.025 is
based on n – 1 = 16 – 1 = 15 degrees of freedom. In
the t distribution table we see that t0.025 = 2.131.
42
Margin of Error and the Interval Estimate:
σ Unknown -4
• Example: Apartment Rents
– Interval Estimate
x ± t0.025
s
n
Margin
of Error
55
750 ± 2.131
=
750 ± 29.30
16
We are 95% confident that the mean rent per month
for the population of one-bedroom apartments within
a half-mile of campus is between $720.70 and $779.30.
43
Margin of Error and the Interval Estimate:
σ Unknown -5
• Example: Consider a study designed to estimate the
mean credit card debt for the population of U.S.
households. A sample of n = 70 households provided
the credit card balances shown in Table.
44
Margin of Error and the Interval Estimate:
σ Unknown -6
• Example:
– For this situation, no previous estimate of the
population standard deviation σ is available.
Thus, the sample data must be used to estimate
both the population mean and the population
standard deviation.
– Using the data, we compute the sample mean x =
$9312 and the sample standard deviation s =
$4007.
45
Margin of Error and the Interval Estimate:
σ Unknown -7
• Example:
– With 95% confidence and n – 1 = 69 degrees of
freedom, t distribution table can be used to obtain
the appropriate value for t0.025.
– We want the t value in the row with 69 degrees of
freedom, and the column corresponding to 0.025
in the upper tail. The value shown is t0.025 = 1.995.
46
Margin of Error and the Interval Estimate:
σ Unknown -8
• Example:
– An interval estimate of the population mean
credit card balance.
9312 ± 1.995
9312 ± 955
4007
70
– The point estimate of the population mean is
$9312, the margin of error is $955, and the 95%
confidence interval is 9312 – 955 = $8357 to 9312 +
955 = $10,267.
– Thus, we are 95% confident that the mean credit
card balance for the population of all households
is between $8357 and $10,267.
47
Using a Small Sample: σ Unknown -1
• Adequate Sample Size
In most applications, a sample size of n = 30 is
adequate when using the expression 𝑥𝑥̅ ± 𝑡𝑡𝛼𝛼⁄2 𝑠𝑠⁄ 𝑛𝑛 to
develop an interval estimate of a population mean.
If the population distribution is highly skewed or
contains outliers, a sample size of 50 or more is
recommended.
48
Using a Small Sample: σ Unknown -2
• Adequate Sample Size (continued)
If the population is not normally distributed but is
roughly symmetric, a sample size as small as 15
will suffice.
If the population is believed to be at least
approximately normal, a sample size of less than 15
can be used.
49
Using a Small Sample: σ Unknown -3
• Example: A sample of 20 employees is
selected, with each employee in the sample
completing the training program. Data on the
training time in days for the 20 employees
are shown in Table.
50
Using a Small Sample: σ Unknown -4
• Example:
– First, the sample data do not support the
conclusion that the distribution of the population
is normal, yet we do not see any evidence of
skewness or outliers. Therefore, using the
guidelines in the previous subsection, we
conclude that an interval estimate based on the t
distribution appears acceptable for the sample of
20 employees.
51
Using a Small Sample: σ Unknown -5
• Example:
– We continue by computing the sample mean and
sample standard deviation as follows.
x
=
s
=
x
∑
=
i
n
1030
= 51.5 days
20
∑ ( xi − x )
2
=
n−1
889
= 6.84 days
20 − 1
52
Using a Small Sample: σ Unknown -6
• Example: Histogram of Training Times for the
Scheer Industries Sample
53
Using a Small Sample: σ Unknown -7
• Example:
– For a 95% confidence interval, we use t
distribution table and n – 1 = 19 degrees of
freedom to obtain t0.025 = 2.093. The interval
estimate of the population mean
 6.84 
51.5 ± 2.093 

 20 
51.5 ± 3.2
– The point estimate of the population mean is 51.5
days. The margin of error is 3.2 days and the 95%
confidence interval is 51.5 – 3.2 = 48.3 days to 51.5
+ 3.2 = 54.7 days.
54
Summary of Interval Estimation
Procedures for a Population Mean
Can the
population standard
deviation σ be assumed
known ?
Yes
Use the sample
standard deviation
s to estimate 
σ Known
Case
Use
x ± zα /2
σ
n
No
σ Unknown
Case
Use
x ± tα /2
s
n
55
8.3 Determining the Sample Size
• Sample Size for an Interval Estimate
of a Population Mean
Let E = the desired margin of error.
E is the amount added to and subtracted from the
point estimate to obtain an interval estimate.
If a desired margin of error is selected prior to
sampling, the sample size necessary to satisfy the
margin of error can be determined.
56
Sample Size for an Interval Estimate
of a Population Mean -1
• Margin of Error
E = zα /2
σ
n
• Necessary Sample Size
( zα / 2 ) 2 σ 2
n=
E2
57
Sample Size for an Interval Estimate
of a Population Mean -2
The Necessary Sample Size equation requires a
value for the population standard deviation σ .
If σ is unknown, a preliminary or planning value
for σ can be used in the equation.
1. Use the estimate of the population standard
deviation computed in a previous study.
2. Use a pilot study to select a preliminary study and
use the sample standard deviation from the study.
3. Use judgment or a “best guess” for the value of σ .
58
Sample Size for an Interval Estimate
of a Population Mean -3
• Example:
– A previous study that investigated the cost of
renting automobiles in the United States found a
mean cost of approximately $55 per day for
renting a midsize automobile.
– Suppose that the organization that conduces this
study would like to conduce a new study in order
to estimate the population mean daily rental cost
for a midsize automobile in the United States.
– In design the new study, the project director
specifies that the population mean daily rental
cost be estimated with a margin of error of $2 and
a 95% level of confidence.
59
Sample Size for an Interval Estimate
of a Population Mean -4
• Example:
– The project director specified a desire margin of
error of E = 2, and the 95% level if confidence
indicates z0.025 = 1.96.
– At this point, we found σ = $9.65, we obtain
( zα /2 )2 σ 2 (1.96)2 (9.65)2
=
n
=
E2
=
89.43
2
2
– The sample size of the new study needs to be at
least 98.43 midsize automobile rentals in order to
satisfy the project director’s $2 margin-of-error
requirement. We round up to the next integer
value, the recommended sample size is 90
midsize automobile rentals.
60
Sample Size for an Interval Estimate
of a Population Mean -5
• Example: Discount Sounds
– Recall that Discount Sounds is evaluating a
potential location for a new retail outlet, based in
part, on the mean annual income of the
individuals in the marketing area of the new
location.
– Suppose that Discount Sounds’ management
team wants an estimate of the population mean
such that there is a 0.95 probability that the
sampling error is $500 or less.
– How large a sample size is needed to meet the
required precision?
61
Sample Size for an Interval Estimate
of a Population Mean -6
• Example: Discount Sounds
zα /2
σ
= 500
n
– At 95% confidence, z0.025 = 1.96. Recall that σ =
4,500.
(1.96)2 (4, 500)2
=
n
=
311.17
=
2
(500)
312
A sample of size 312 is needed to reach a desired
precision of ± $500 at 95% confidence.
62
8.4 Population Proportion
• Determining the Sample Size
63
Interval Estimate
of a Population Proportion -1
The general form of an interval estimate of a
population proportion p is
p ± Margin of Error
64
Interval Estimate
of a Population Proportion -2
The sampling distribution of plays a key role in
computing the margin of error for this interval
estimate.
The sampling distribution of can be approximated
by a normal distribution whenever np > 5 and
n(1 – p) > 5.
65
Interval Estimate
of a Population Proportion -3
• Normal Approximation of Sampling
Distribution of p
Sampling
distribution
of p
α/2
p(1 − p)
σp =
n
1 - α of all
p values
zα /2σ p
p
α/2
p
zα /2σ p
66
Interval Estimate
of a Population Proportion -4
• Interval Estimate
p ± zα / 2
p (1 − p )
n
where:
1 – α is the confidence coefficient
zα/2 is the z value providing an area of α/2 in
the upper tail of the standard normal
probability distribution
p is the sample proportion
67
Interval Estimate
of a Population Proportion -5
• Example: Political Science, Inc.
– Political Science, Inc. (PSI) specializes in voter
polls and surveys designed to keep political office
seekers informed of their position in a race.
– Using telephone surveys, PSI interviewers ask
registered voters who they would vote for if the
election were held that day.
68
Interval Estimate
of a Population Proportion -6
• Example: Political Science, Inc.
– In a current election campaign, PSI has just found
that 220 registered voters, out of 500 contacted,
favor a particular candidate. PSI wants to develop
a 95% confidence interval estimate for the
proportion of the population of registered voters
that favor the candidate.
69
Interval Estimate
of a Population Proportion -7
• Example: Political Science, Inc.
p ± zα /2
– where:
p(1 − p )
n
n = 500, p = 220/500 = 0.44,
zα/2 = 1.96
0.44(1 − 0.44)
0.44 ± 1.96
500
PSI is 95% confident that the proportion of all voters
that favor the candidate is between 0.3965 and 0.4835.
70
Interval Estimate
of a Population Proportion -8
• Example: Tee Times
– A national survey of 900 women golfers was
conduced to learn how women golfers view their
treatment at golf courses in the United States. The
survey found that 396 of the women golfers were
satisfied with the availability of tee times.
– The point estimate of the proportion of the
population of women golfers who are satisfied
with the availability of tee times is 396/900 = 0.44.
71
Interval Estimate
of a Population Proportion -9
• Example: Tee Times
– A 95% confidence level, p ± zα /2
p(1 − p )
n
0.44(1 − 0.44)
0.44 ± 1.96
900
0.44 ± 0.0324
– The margin of error us 0.0324 and the 95%
confidence interval estimate of the population
proportion is 0.4074 to 0.4724. The survey results
enable us to state with 95% confidence that
between 40.76% to 47.24% of all women golfers
are satisfied with the availability of tee times.
72
Sample Size for an Interval Estimate
of a Population Proportion -1
• Margin of Error
E = zα / 2
p (1 − p )
n
– Solving for the necessary sample size, we get
( zα /2 ) 2 p (1 − p )
n=
E2
– However, p will not be known until after we
have selected the sample. We will use the
planning value p* for p .
73
Sample Size for an Interval Estimate
of a Population Proportion -2
• Necessary Sample Size
( zα / 2 ) 2 p* (1 − p* )
n=
E2
• The planning value p* can be chosen by:
1. Using the sample proportion from a previous
sample of the same or similar units, or
2. Selecting a preliminary sample and using the
sample proportion from this sample.
3. Use judgment or a “best guess” for a p* value.
4. Otherwise, use 0.50 as the p* value.
74
Sample Size for an Interval Estimate
of a Population Proportion -3
• Some Possible Values for p*(1 − p*)
75
抽取1,000份樣本的原因
• 民意、市場調查的多為封閉問卷，有興
趣的多為某個問項佔的比例，例如：某
位候選人的支持程度二項分配。
• 在信心水準為95%及最大誤差不大於 3%
的要求下：
1.96 ×
p (1 − p )
≤ 0.03
n
1.96 p (1 − p ) 1.96 × 1 / 2
⇔ n≥
≅
0.03
0.03
⇔ n ≥ 1,067
Sample Size for an Interval Estimate
of a Population Proportion -4
• Example: Political Science, Inc.
– Suppose that PSI would like a 0.99 probability
that the sample proportion is within + 0.03 of the
population proportion.
– How large a sample size is needed to meet the
required precision? (A previous sample of similar
units yielded 0.44 for the sample proportion.)
77
Sample Size for an Interval Estimate
of a Population Proportion -5
• Example: Political Science, Inc.
p(1 − p )
zα /2
= 0.03
n
– At 99% confidence, z0.005 = 2.576. Recall that p =
0.44.
( zα /2 )2 p * (1 − p *) (2.576)2 (0.44)(0.56)
n =
≅ 1817
2
2
(0.03)
E
A sample of size 1817 is needed to reach a desired
precision of ± 0.03 at 99% confidence.
78
Sample Size for an Interval Estimate
of a Population Proportion -6
• Example: Political Science, Inc.
– Note: We used 0.44 as the best estimate of p in the
preceding expression. If no information is
available about p, then 0.5 is often assumed
because it provides the highest possible sample
size. If we had used p = 0.5, the recommended n
would have been 1843.
79
Sample Size for an Interval Estimate
of a Population Proportion -7
• Example: Tee Times
– How large should the sample be if the survey
director wants to estimate the population
proportion with a margin of error of 0.25 at 95%
confidence?
– With E = 0.25 and zα/2 = 1.96, we need a planning
value p* to answer the sample size question.
80
Sample Size for an Interval Estimate
of a Population Proportion -8
• Example: Tee Times
– Using the previous survey result of p = 0.44 as the
planning value p*,
n
( zα /2 )2 p * (1 − p *) (1.96)2 (0.44)(1 − 0.44)
1514.5
=
=
2
2
(0.025)
E
– The sample size must be at least 1514.5 women
golfers to satisfy the margin of error requirement.
Rounding up to the next integer value indicates
that a sample of 1515 women golfers is
recommended to satisfy the margin of error
requirement.
81
Sample Size for an Interval Estimate
of a Population Proportion -9
• Example: Tee Times
– The fourth alternative suggested for selecting a
planning value p* is to use p* = 0.50. This value of
p* is frequently used when no other information
is available.
– In the survey of women golfers example, a
planning value of p* = 0.50 would have provided
the sample size
( zα /2 )2 p * (1 − p *) (1.96)2 (0.50)(1 − 0.50)
n =
1536.6
=
=
2
2
(0.025)
E
Thus, a slightly larger sample size of 1537 women
golfers would be recommended.
82
End of Chapter 8
83