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Slides Prepared by
JOHN S. LOUCKS
St. Edward’s University
© 2003 South-Western /Thomson Learning™
Slide 1
Chapter 8
Interval Estimation




Interval Estimation of a Population Mean:
Large-Sample Case
Interval Estimation of a Population Mean:
Small-Sample Case
Determining the Sample Size to Estimate 
Interval Estimation of a Population Proportion

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© 2003 South-Western /Thomson Learning™
Slide 2
Interval Estimation of a Population Mean:
Large-Sample Case




Margin of Error and the Interval Estimate
Probability Statements about the Sampling Error
Constructing an Interval Estimate:
Large-Sample Case with  Assumed Known
Calculating an Interval Estimate:
Large-Sample Case with  Estimated by s
© 2003 South-Western /Thomson Learning™
Slide 3
Margin of Error and the Interval Estimate



A point estimator does not provide information about
how close the estimate is to the population parameter.
For this reason statisticians usually prefer to construct
an interval estimate.
An interval estimate is constructed by subtracting and
adding a value, called the margin of error, to a point
estimate.
Point Estimate  Margin of Error
© 2003 South-Western /Thomson Learning™
Slide 4
Margin of Error and the Interval Estimate

An interval estimate of the population mean is:
x  Margin of Error

An interval estimate of the population proportion is:
p  Margin of Error
© 2003 South-Western /Thomson Learning™
Slide 5
Sampling Error


The absolute value of the difference between an
unbiased point estimate and the population
parameter it estimates is called the sampling error.
For the case of a sample mean estimating a
population mean:
Sampling Error = | x  |
© 2003 South-Western /Thomson Learning™
Slide 6
Probability Statements
About the Sampling Error


Knowledge of the sampling distribution of x enables
us to make probability statements about the
sampling error even though the population mean 
is not known.
A probability statement about the sampling error is a
precision statement.
© 2003 South-Western /Thomson Learning™
Slide 7
Probability Statements
About the Sampling Error

Precision Statement
There is a 1 -  probability that the value of a
sample mean will provide a sampling error of
z /2  x or less.
Sampling
distribution
of x
1 -  of all
/2
/2
x values

© 2003 South-Western /Thomson Learning™
x
Slide 8
Interval Estimate of a Population Mean:
Large-Sample Case (n > 30)

With  Assumed Known
x  z /2
where:

n
x is the sample mean
1 - is the confidence coefficient
z/2 is the z value providing an area of
/2 in the upper tail of the standard
normal probability distribution
 is the population standard deviation
n is the sample size
© 2003 South-Western /Thomson Learning™
Slide 9
Interval Estimate of a Population Mean:
Large-Sample Case (n > 30)

With  Estimated by s
In most applications the value of the population
standard deviation is unknown. We simply use the
value of the sample standard deviation, s, as the
point estimate of the population standard deviation.
x  z /2
© 2003 South-Western /Thomson Learning™
s
n
Slide 10
Example: National Discount, Inc.
National Discount has 260 retail outlets
throughout the United States. National evaluates
each potential location for a new retail outlet in part
on the mean annual income of the individuals in the
marketing area of the new location.
Sampling can be used to develop an interval
estimate of the mean annual income for individuals
in a potential marketing area for National Discount.
A sample of size n = 36 was taken. The sample
mean, x , is $21,100 and the sample standard
deviation, s, is $4,500. We will use .95 as the
confidence coefficient in our interval estimate.
© 2003 South-Western /Thomson Learning™
Slide 11
Example: National Discount, Inc.

Precision Statement
There is a .95 probability that the value of a
sample mean for National Discount will provide a
sampling error of $1,470 or less……. determined as
follows:
95% of the sample means that can be observed
are within + 1.96  x of the population mean .
If  x  s
n
 4,500
© 2003 South-Western /Thomson Learning™
36
 750 , then 1.96  x = 1,470.
Slide 12
Example: National Discount, Inc.

Interval Estimate of the Population Mean:  Unknown
Interval Estimate of  is:
$21,100 + $1,470
or $19,630 to $22,570
We are 95% confident that the interval contains the
population mean.
© 2003 South-Western /Thomson Learning™
Slide 13
Using Excel to Construct
a Confidence Interval: Large-Sample Case

Formula Worksheet
1
2
3
4
5
6
7
8
9
10
11
12
13
14
A
Income
25,600
19,615
20,035
21,735
17,600
26,080
28,925
25,350
15,900
17,550
14,745
20,000
19,250
B
C
Sample Size 36
Sample Mean =AVERAGE(A2:A37)
Sample Std. Deviation =STDEV(A2:A37)
Confidence Coefficient 0.95
Level of Signif. (alpha) =1-C5
z Value =NORMSINV(1-C5/2)
Standard Error =C3/SQRT(C1)
Margin of Error =C7*C9
Point Estimate =C2
Lower Limit =C12-C10
Upper Limit =C12+C10
Note: Rows 15-37 are not shown.
© 2003 South-Western /Thomson Learning™
Slide 14
Using Excel to Construct
a Confidence Interval: Large-Sample Case

Value Worksheet
1
2
3
4
5
6
7
8
9
10
11
12
13
14
A
Income
25,600
19,615
20,035
21,735
17,600
26,080
28,925
25,350
15,900
17,550
14,745
20,000
19,250
B
Sample Size 36
Sample Mean 21,100
Sample Std. Deviation 4500
C
Confidence Coefficient 0.95
Level of Signif. (alpha) 0.05
z Value 1.960
Standard Error 749.992
Margin of Error 1469.955
Point Estimate 21,100.39
Lower Limit 19,630.43
Upper Limit 22,570.34
Note: Rows 15-37 are not shown.
© 2003 South-Western /Thomson Learning™
Slide 15
Interval Estimation of a Population Mean:
Small-Sample Case (n < 30)



Population is Not Normally Distributed
The only option is to increase the sample size to
n > 30 and use the large-sample interval-estimation
procedures.
Population is Normally Distributed and  is Assumed
Known
The large-sample interval-estimation procedure can
be used.
Population is Normally Distributed and  is Estimated
by s
The appropriate interval estimate is based on a
probability distribution known as the t distribution.
© 2003 South-Western /Thomson Learning™
Slide 16
t Distribution





The t distribution is a family of similar probability
distributions.
A specific t distribution depends on a parameter
known as the degrees of freedom.
As the number of degrees of freedom increases, the
difference between the t distribution and the
standard normal probability distribution becomes
smaller and smaller.
A t distribution with more degrees of freedom has
less dispersion.
The mean of the t distribution is zero.
© 2003 South-Western /Thomson Learning™
Slide 17
Interval Estimation of a Population Mean:
Small-Sample Case (n < 30) with  Estimated by s

Interval Estimate
x  t /2
where
s
n
1 - = the confidence coefficient
t/2 = the t value providing an area of /2
in the upper tail of a t distribution
with n - 1 degrees of freedom
s = the sample standard deviation
© 2003 South-Western /Thomson Learning™
Slide 18
Example: Apartment Rents

Interval Estimation of a Population Mean:
Small-Sample Case (n < 30) with  Estimated by s
A reporter for a student newspaper is writing an
article on the cost of off-campus housing. A sample
of 10 one-bedroom units within a half-mile of
campus resulted in a sample mean of $550 per month
and a sample standard deviation of $60.
Let us provide a 95% confidence interval
estimate of the mean rent per month for the
population of one-bedroom units within a half-mile
of campus. We’ll assume this population to be
normally distributed.
© 2003 South-Western /Thomson Learning™
Slide 19
Example: Apartment Rents

t Value
At 95% confidence, 1 -  = .95,  = .05, and /2 = .025.
t.025 is based on n - 1 = 10 - 1 = 9 degrees of freedom.
In the t distribution table we see that t.025 = 2.262.
Degrees
Area in Upper Tail
of Freedom
.10
.05
.025
.01
.005
.
.
.
.
.
.
7
1.415
1.895
2.365
2.998
3.499
8
1.397
1.860
2.306
2.896
3.355
9
1.383
1.833
2.262
2.821
3.250
10
1.372
1.812
2.228
2.764
3.169
.
.
.
.
.
.
© 2003 South-Western /Thomson Learning™
Slide 20
Example: Apartment Rents

Interval Estimation of a Population Mean:
Small-Sample Case (n < 30) with  Estimated by s
x  t.025
s
n
60
10
550 + 42.92
$507.08 to $592.92
550  2.262
or
We are 95% confident that the mean rent per month
for the population of one-bedroom units within a
half-mile of campus is between $507.08 and $592.92.
© 2003 South-Western /Thomson Learning™
Slide 21
Using Excel to Construct
a Confidence Interval:  Estimated by s

Formula Worksheet
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A
Rent
600
635
550
535
465
625
570
465
550
505
B
C
Sample Size 10
Sample Mean =AVERAGE(A2:A11)
Sample Std. Deviation =STDEV(A2:A11)
Confidence Coefficient
Level of Signif. (alpha)
Degrees of Freedom
t Value
0.95
=1-C5
=C1-1
=TINV(C6,C7)
Standard Error =C3/SQRT(C1)
Margin of Error =C8*C10
© 2003 South-Western /Thomson Learning™
Point Estimate =C2
Lower Limit =C13-C11
Upper Limit =C13+C11
Slide 22
Using Excel to Construct
a Confidence Interval:  Estimated by s

Value Worksheet
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A
Rent
600
635
550
535
465
625
570
465
550
505
B
Sample Size 10
Sample Mean 550.00
Sample Std. Deviation 60.05
Confidence Coefficient
Level of Signif. (alpha)
Degrees of Freedom
t Value
C
0.95
0.05
9
2.2622
Standard Error 18.988
Margin of Error 42.955
© 2003 South-Western /Thomson Learning™
Point Estimate 550.00
Lower Limit 507.05
Upper Limit 592.95
Slide 23
Sample Size for an Interval Estimate
of a Population Mean





Let E = the maximum sampling error mentioned in
the precision statement.
E is the amount added to and subtracted from the
point estimate to obtain an interval estimate.
E is often referred to as the margin of error.
We have

E  z /2
n
Solving for n we have
( z / 2 ) 2  2
n
E2
© 2003 South-Western /Thomson Learning™
Slide 24
Example: National Discount, Inc.

Sample Size for an Interval Estimate of a Population
Mean
Suppose that National’s management team
wants an estimate of the population mean such that
there is a .95 probability that the sampling error is
$500 or less.
How large a sample size is needed to meet the
required precision?
© 2003 South-Western /Thomson Learning™
Slide 25
Example: National Discount, Inc.

Sample Size for Interval Estimate of a Population Mean
z /2

 500
n
At 95% confidence, z.025 = 1.96.
Recall that = 4,500.
Solving for n we have
(1.96)2 (4,500)2
n
 311.17
2
(500)
We need to sample 312 to reach a desired precision of
+ $500 at 95% confidence.
© 2003 South-Western /Thomson Learning™
Slide 26
Interval Estimation
of a Population Proportion

Interval Estimate
p  z / 2
where:
p (1  p )
n
1 - is the confidence coefficient
z/2 is the z value providing an area of
/2 in the upper tail of the standard
normal probability distribution
p is the sample proportion
© 2003 South-Western /Thomson Learning™
Slide 27
Example: Political Science, Inc.

Interval Estimation of a Population Proportion
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, interviewers ask registered voters
who they would vote for if the election were held
that day.
In a recent election campaign, PSI found that 220
registered voters, out of 500 contacted, favored a
particular candidate. PSI wants to develop a 95%
confidence interval estimate for the proportion of the
population of registered voters that favors the
candidate.
© 2003 South-Western /Thomson Learning™
Slide 28
Example: Political Science, Inc.

Interval Estimate of a Population Proportion
p  z / 2
p (1  p )
n
where: n = 500, p = 220/500 = .44, z/2 = 1.96
. 44(1. 44)
. 44  1. 96
500
.44 + .0435
PSI is 95% confident that the proportion of all voters
that favors the candidate is between .3965 and .4835.
© 2003 South-Western /Thomson Learning™
Slide 29
Using Excel to Construct a Confidence Interval

Formula Worksheet
1
2
3
4
5
6
7
8
9
10
11
12
13
14
B
A
Sample Size
Favored
Total Yes
Yes
Sample Proportion
Yes
No
Confid. Coefficient
Yes
Lev. of Signif. (alpha)
No
z Value
No
No
Standard Error
No
Margin of Error
Yes
No
Point Estimate
Yes
Lower Limit
No
Upper Limit
No
C
500
=COUNTIF(A2:A501,"Yes")
=C2/C1
0.95
=1-C5
=NORMSINV(1-C6/2)
=SQRT(C3*(1-C3)/C1)
=C7*C9
=C3
=C12-C10
=C12+C10
Note: Rows 15-501 are not shown.
© 2003 South-Western /Thomson Learning™
Slide 30
Using Excel to Construct a Confidence Interval

Value Worksheet
1
2
3
4
5
6
7
8
9
10
11
12
13
14
A
B
Favored
Sample Size
Yes
Total Yes
Yes
Sample Proportion
No
Yes
Confid. Coefficient
No
Lev. of Signif. (alpha)
No
z Value
No
No
Standard Error
Yes
Margin of Error
No
Yes
Point Estimate
No
Lower Limit
No
Upper Limit
C
500
220
0.4400
0.95
0.05
1.9600
0.02220
0.04351
0.4400
0.3965
0.4835
Note: Rows 15-501 are not shown.
© 2003 South-Western /Thomson Learning™
Slide 31
Sample Size for an Interval Estimate
of a Population Proportion

Let E = the maximum sampling error mentioned in
the precision statement.
We have
p(1  p)
E  z / 2
n

Solving for n we have

( z / 2 ) 2 p(1  p)
n
E2
© 2003 South-Western /Thomson Learning™
Slide 32
Example: Political Science, Inc.

Sample Size for an Interval Estimate of a Population
Proportion
Suppose that PSI would like a .99 probability that
the sample proportion is within + .03 of the
population proportion.
How large a sample size is needed to meet the
required precision?
© 2003 South-Western /Thomson Learning™
Slide 33
Example: Political Science, Inc.

Sample Size for Interval Estimate of a Population
Proportion
At 99% confidence, z.005 = 2.576.
( z / 2 ) 2 p(1  p) ( 2.576) 2 (. 44)(.56)
n

 1817
2
2
E
(. 03)
Note: We used .44 as the best estimate of p in the
above expression. If no information is available
about p, then .5 is often assumed because it provides
the highest possible sample size. If we had used
p = .5, the recommended n would have been 1843.
© 2003 South-Western /Thomson Learning™
Slide 34
End of Chapter 8
© 2003 South-Western /Thomson Learning™
Slide 35