Chapter 8: Statistical Inference: Estimation for Single Populations
Chapter 8
Statistical Inference: Estimation for Single Populations
LEARNING OBJECTIVES
The overall learning objective of Chapter 8 is to help you understand estimating
parameters of single populations, thereby enabling you to:
1.
2.
3.
4.
5.
6.
Know the difference between point and interval estimation.
Estimate a population mean from a sample mean when  is known.
Estimate a population mean from a sample mean when  is unknown.
Estimate a population proportion from a sample proportion.
Estimate the population variance from a sample variance.
Estimate the minimum sample size necessary to achieve given statistical
goals.
CHAPTER TEACHING STRATEGY
Chapter 8 is the student's introduction to interval estimation and
estimation of sample size. In this chapter, the concept of point estimate is
discussed along with the notion that as each sample changes in all likelihood so
will the point estimate. From this, the student can see that an interval estimate
may be more usable as a one-time proposition than the point estimate. The
confidence interval formulas for large sample means and proportions can be
presented as mere algebraic manipulations of formulas developed in chapter 7
from the Central Limit Theorem.
It is very important that students begin to understand the difference
between mean and proportions. Means can be generated by averaging some sort
of measurable item such as age, sales, volume, test score, etc. Proportions are
computed by counting the number of items containing a characteristic of interest
out of the total number of items. Examples might be proportion of people
carrying a VISA card, proportion of items that are defective, proportion of market
purchasing brand A. In addition, students can begin to see that sometimes single
samples are taken and analyzed; but that other times, two samples are taken in
order to compare two brands, two techniques, two conditions, male/female, etc.
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Chapter 8: Statistical Inference: Estimation for Single Populations
In an effort to understand the impact of variables on confidence intervals,
it may be useful to ask the students what would happen to a confidence interval if
the sample size is varied or the confidence is increased or decreased. Such
consideration helps the student see in a different light the items that make up a
confidence interval. The student can see that increasing the sample size reduces
the width of the confidence interval, all other things being constant, or that it
increases confidence if other things are held constant. Business students probably
understand that increasing sample size costs more and thus there are trade-offs in
the research set-up.
In addition, it is probably worthwhile to have some discussion with
students regarding the meaning of confidence, say 95%. The idea is presented in
the chapter that if 100 samples are randomly taken from a population and 95%
confidence intervals are computed on each sample, that 95%(100) or 95 intervals
should contain the parameter of estimation and approximately 5 will not. In most
cases, only one confidence interval is computed, not 100, so the 95% confidence
puts the odds in the researcher's favour. It should be pointed out, however, that
the confidence interval computed may not contain the parameter of interest.
This chapter introduces the student to the t distribution for estimating
population means when  is unknown. Emphasize that this applies only when the
population is normally distributed because it is an assumption underlying the t test
that the population is normally distributed albeit that this assumption is robust.
The student will observe that the t formula is essentially the same as the z formula
and that it is the table that is different. When the population is normally
distributed and  is known, the z formula can be used even for small samples.
A formula is given in chapter 8 for estimating the population variance; and
it is here that the student is introduced to the chi-square distribution. An
assumption underlying the use of this technique is that the population is normally
distributed. The use of the chi-square statistic to estimate the population variance
is extremely sensitive to violations of this assumption. For this reason, extreme
caution should be exercised in using this technique. Because of this, some
statisticians omit this technique from consideration presentation and usage.
Lastly, this chapter contains a section on the estimation of sample size.
One of the more common questions asked of statisticians is: "How large of a
sample size should I take?" In this section, it should be emphasized that sample
size estimation gives the researcher a "ball park" figure as to how many to sample.
The “error of estimation" is a measure of the sampling error. It is also equal to
the + error of the interval shown earlier in the chapter.
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Chapter 8: Statistical Inference: Estimation for Single Populations
CHAPTER OUTLINE
8.1
Estimating the Population Mean Using the z Statistic ( known).
Finite Correction Factor
Estimating the Population Mean Using the z Statistic when the
Sample Size is Small
Using the Computer to Construct z Confidence Intervals for the
Mean
8.2
Estimating the Population Mean Using the t Statistic ( unknown).
The t Distribution
Robustness
Characteristics of the t Distribution.
Reading the t Distribution Table
Confidence Intervals to Estimate the Population Mean Using the t
Statistic
Using the Computer to Construct t Confidence Intervals for the
Mean
8.3
Estimating the Population Proportion
8.4
Estimating the Population Variance
8.5
Estimating Sample Size
Determining Sample Size When Estimating µ
Determining Sample Size When Estimating p
KEY WORDS
Chi-square Distribution
Degrees of Freedom(df)
Error of Estimation (or error of the interval)
Interval Estimate
Interval Estimate
Point Estimate
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Precision (or bounds)
Robust
Sample-Size Estimation
t Distribution
t Value
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Chapter 8: Statistical Inference: Estimation for Single Populations
SOLUTIONS TO PROBLEMS IN CHAPTER 8
8.1 a)
 = 3.5
x = 25
95% Confidence
xz
b)

n
x = 3.419
90% C.I.
xz
d)
n
= 25 + 1.96
x = 119.6
98% Confidence
xz
c)


n
x = 56.7
80% C.I.
xz

n
n = 60
z.025 = 1.96
3 .5
= 25 + 0.89 = 24.11 < µ < 25.89
60
 = 23.89
n = 75
z.01 = 2.33
= 119.6 + 2.33
23.89
75
 = 0.974
= 3.419 + 1.645
= 119.6 ± 6.43 = 113.17 < µ < 126.03
n = 32
z.05 = 1.645
0.974
= 3.419 ± .283 = 3.136 < µ < 3.702
32
 = 12.1
N = 500
n = 47
z.10 = 1.28
12.1 500  47
N n
= 56.7 + 1.28
N 1
47 500  1
=
56.7 ± 2.15 = 54.55 < µ < 58.85
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.2 n = 36
95% C.I.
xz

n
8.3 n = 81
90% C.I.

xz
n
x = 90.4
94% C.I.

n
8.5 n = 39
96% C.I.
xz
= 211 ± 1.96
23
36

n
=
211 ± 7.51 = 203.49 < µ < 218.51
 = 5.89
x = 47
z.05=1.645
= 47 ± 1.645
5.89
= 47 ± 1.08 = 45.92 < µ < 48.08
81
2 = 49
8.4 n = 70
xz
 = 23
x = 211
z.025 = 1.96
x = 90.4
Point Estimate
z.03 = 1.88
= 90.4 ± 1.88
49
= 90.4 ± 1.57 = 88.83 < µ < 91.97
70
N = 200
z.02 = 2.05
 = 11
x = 66
N n
11
= 66 ± 2.05
N 1
39
200  39
=
200  1
66 ± 3.25 = 62.75 < µ < 69.25
x = 66
Point Estimate
8.6 n = 120
99% C.I.
 = 0.8735
x = 18.72
z.005 = 2.575
x = 18.72 Point Estimate
xz

n
= 18.72 ± 2.575
0.8735
= 18.72 ± .21 = 18.51 < µ < 18.93
120
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.7 N = 1500
95% C.I.
n = 187
z.025 = 1.96
x = 5.3 years

xz
n
x = 5.3 years
 = 1.28 years
Point Estimate
N n
1.28 1500  187
= 5.3 ± 1.96
=
N 1
1500  1
187
5.3 ± .17 = 5.13 < µ < 5.47
8.8 n = 24
90% C.I.
xz

n
8.9 n = 36
98% C.I.
xz

n
8.10 n = 36
 = 3.23
x = 5.625
z.05 = 1.645
= 5.625 ± 1.645
3.23
24
 = 1.17
x = 3.306
z.01 = 2.33
= 3.306 ± 2.33
= 5.625 ± 1.085 = 4.540 < µ < 6.710
1.17
36
= 3.306 ± .454 = 2.852 < µ < 3.760
 = .113
x = 2.139
x = 2.139 Point Estimate
90% C.I.
xz

n
z.05 = 1.645
= 2.139 ± 1.645
(.113)
= 2.139 ± .031 = 2.108 < µ < 2.170
36
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.11 95% confidence interval
x = 24.511
xz

n
n = 45
= 5.124
= 24.511 + 1.96
z.025 = 1.96
5.124
45
=
24.511 + 1.497 = 23.014 <  < 26.008
x = 24.511 Point Estimate
Error of the Interval = 1.497
8.12 The point estimate is 0.5765.
The assumed standard deviation is 0.140000.
Sample size:
n = 41
Error of the estimate (error of the interval):

0.140000
z 2
 z 2
 0.042853
n
41
z 2 
0.619353 – 0.576500 = 0.042853
0.042853 41
 1.96
0.140000
95% level of confidence
Confidence interval:
8.13 n = 13
0.533647 < µ < 0.619353
x = 45.62
s = 5.694
df = 13 – 1 = 12
95% Confidence Interval and /2=.025
t.025,12 = 2.179
xt
s
n
8.14 n = 12
= 45.62 ± 2.179
5.694
= 45.62 ± 3.44 = 42.18 < µ < 49.06
13
x = 319.17
s = 9.104
df = 12 – 1 = 11
90% confidence interval
/2 = .05
t.05,11 = 1.796
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Chapter 8: Statistical Inference: Estimation for Single Populations
xt
s
n
= 319.17 ± (1.796)
8.15 n = 41
9.104
= 319.17 ± 4.72 = 314.45 < µ < 323.89
12
x = 128.4
s = 20.6
df = 41 – 1 = 40
98% Confidence Interval
/2 = .01
t.01,40 = 2.423
xt
s
n
= 128.4 ± 2.423
20.6
41
= 128.4 ± 7.80 = 120.6 < µ < 136.2
x = 128.4 Point Estimate
8.16 n = 15
s2 = 0.81
x = 2.364
df = 15 – 1 = 14
90% Confidence interval
/2 = .05
t.05,14 = 1.761
xt
s
n
= 2.364 ± 1.761
8.17 n = 25
0.81
= 2.364 ± .409 = 1.955 < µ < 2.773
15
x = 16.088
s = .817
df = 25 – 1 = 24
99% Confidence Interval
/2 = .005
t.005,24 = 2.797
xt
s
n
= 16.088 ± 2.797
(.817 )
25
= 16.088 ± .457 = 15.631 < µ < 16.545
x = 16.088 Point Estimate
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.18 n = 51
x = 1,192
s = 279
98% CI and /2 = .01
xt
s
n
df = n – 1 = 50
t.01,50 = 2.403
= 1,192 + 2.403
279
51
= 1,192 + 93.880 = 1,098.12 <  < 1,285.88
The figure given by Runzheimer International falls within the confidence
interval. Therefore, there is no reason to reject the Runzheimer figure as
different from what we are getting based on this sample.
8.19 n = 20
df = 19
x = 2.36116
95% CI
t.025,19 = 2.093
s = 0.19721
2.36116 + 2.093
0.1972
= 2.36116 + 0.0923 = 2.26886 <  < 2.45346
20
Point Estimate = 2.36116
Error = 0.0923
8.20 n = 28
x = 5.264
s = 2.043
df = 28 – 1 = 27
/2 = .05
90% Confidence Interval
t.05,27 = 1.703
xt
s
n
8.21 n = 10
= 5.264 ± 1.703
x = 49.8
95% Confidence
xt
s
n
2.043
28
s = 18.22
/2 = .025
= 49.8 ± 2.262
= 5.264 + .658 = 4.606 < µ < 5.922
df = 10 – 1 = 9
t.025,9 = 2.262
18.22
= 49.8 + 13.03 = 36.77 < µ < 62.83
10
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.22 n = 14
/2 = .01
98% confidence
df = 13
t.01,13 = 2.650
from data:
x = 152.16
confidence interval:
xt
s = 14.42
s
n
= 152.16 + 2.65
14.42
=
14
152.16 + 10.21 = 141.95 <  < 162.37
The point estimate is 152.16
8.23
n = 41
df = 41 – 1 = 40
/2 = .005
99% confidence
t.005,40 = 2.704
from data:
x = 11.10
confidence interval:
xt
s = 8.45
s
n
= 11.10 + 2.704
8.45
41
=
11.10 + 3.57 = 7.53 <  < 14.67
8.24
The point estimate is x which is 25.4134 hours. The sample size is 26 skiffs.
The confidence level is 98%. The confidence interval is:
s
s
x t
  x t
= 22.8124 <  < 28.0145
n
n
The error of the confidence interval is 2.6011.
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.25
a)
n = 44
n = 300
a)
n = 95
pˆ  z
90% C.I.
z.05 = 1.645
p̂ = .32
88% C.I.
z.06 = 1.555
x = 57
99% C.I.
z.005 = 2.575
(.49)(.51)
pˆ  qˆ
= .49 ± 2.575
= .49 ± .12 = .37 < p < .61
n
116
n = 800
p̂ =
p̂ = .48
x
57

= .49
n 116
pˆ  z
b)
z.025 = 1.96
(.32)(.68)
pˆ  qˆ
= .32 ± 1.555
= .32 ± .074 = .246 < p < .394
n
95
n = 116
p̂ =
95% C.I.
(.48)(.52)
pˆ  qˆ
= .48 ± 1.645
= .48 ± .024 = .456 < p < .504
n
1150
pˆ  z
8.26
p̂ = .82
n = 1150
pˆ  z
d)
z.005 = 2.575
(.82)(.18)
pˆ  qˆ
= .82 ± 1.96
= .82 ± .043 = .777 < p < .863
n
300
pˆ  z
c)
99% C.I.
(.51)(.49)
pˆ  qˆ
= .51 ± 2.575
= .51 ± .194 = .316 < p< .704
n
44
pˆ  z
b)
p̂ =.51
x = 479
97% C.I.
z.015 = 2.17
x 479

= .60
n 800
(.60)(.40)
pˆ  qˆ
= .60 ± 2.17
= .60 ± .038 = .562 < p < .638
n
800
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Chapter 8: Statistical Inference: Estimation for Single Populations
c)
n = 240
p̂ =
pˆ  z
8.27
n = 85
p̂ =
pˆ  z
90% C.I.
z.05 = 1.645
x 21

= .35
n 60
(.35)(.65)
pˆ  qˆ
= .35 ± 1.645
= .35 ± .10 = .25 < p < .45
n
60
90% C.I.
z.05 = 1.645
x 40

= .47
n 85
(.47)(.53)
pˆ  qˆ
= .47 ± 1.645
= .47 ± .09 = .38 < p < .56
n
85
z.025 = 1.96
(.47)(.53)
pˆ  qˆ
= .47 ± 1.96
= .47 ± .11 = .36 < p < .58
n
85
99% C.I.
pˆ  z
x = 21
x = 40
95% C.I.
pˆ  z
z.075 = 1.44
(.44)(.56)
pˆ  qˆ
= .44 ± 1.44
= .44 ± .046 = .394 < p < .486
n
240
n = 60
p̂ =
85% C.I.
x 106

= .44
n 240
pˆ  z
d)
x = 106
z.005 = 2.575
(.47)(.53)
pˆ  qˆ
= .47 ± 2.575
= .47 ± .14 = .33 < p < .61
n
85
All other things being constant, as the confidence increased, the width of the
interval increased.
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.28
n = 1003
p̂ = .255
99% CI
z.005 = 2.575
pˆ  qˆ
(.255)(.745)
= .255 + 2.575
= .255 + .035 = .220 < p < .290
n
1003
pˆ  z
n = 10,000
p̂ = .255
99% CI
z.005 = 2.575
(.255)(. 745)
pˆ  qˆ
= .255 + 2.575
= .255 + .011 = .244 < p < .266
10,000
n
pˆ  z
The confidence interval constructed using n = 1003 is wider than the confidence
interval constructed using n = 10,000. One might conclude that, all other things
being constant, increasing the sample size reduces the width of the confidence
interval.
8.29
n = 560
p̂ = .47
n = 560
8.30
p̂ = .28
pˆ  z
90% CI
z.05 = 1.645
(.28)(.72)
pˆ  qˆ
= .28 + 1.645
= .28 + .0312 = .2488 < p < .3112
n
560
n = 1250
p̂ =
z.025 = 1.96
(.47)(.53)
pˆ  qˆ
= .47 + 1.96
= .47 + .0413 = .4287 < p < .5113
n
560
pˆ  z
pˆ  z
95% CI
x = 997
98% C.I.
z.01 = 2.33
x
997

= .80
n 1250
(.80)(.20)
pˆ  qˆ
= .80 ± 2.33
= .80 ± .026 = .774 < p < .826
n
1250
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.31
n = 3481
p̂ =
x = 927
x
927

= .266
n 3481
a)
p̂ = .266 Point Estimate
b)
99% C.I.
pˆ  z
z.005 = 2.575
(.266)(.734)
pˆ  qˆ
= .266 + 2.575
= .266 ± .019 =
n
3481
.247 < p < .285
8.32
n = 89
p̂ =
p̂ = .63
pˆ  z
n = 672
95% Confidence
z.025 = 1.96
(.63)(.37)
pˆ  qˆ
= .63 + 1.96
= .63 + .0365 = .5935 < p < .6665
n
672
n = 275
p̂ =
z.075 = 1.44
(.54)(.46)
pˆ  qˆ
= .54 ± 1.44
= .54 ± .076 = .464 < p < .616
n
89
pˆ  z
8.34
85% C.I.
x 48

= .54
n 89
pˆ  z
8.33
x = 48
x = 121
98% confidence
z.01 = 2.33
x 121

= .44
n 275
pˆ  qˆ
(.44)(.56)
= .44 ± 2.33
= .44 ± .07 = .37 < p < .51
n
275
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.35
a)
n = 12
s2 = 44.9
x = 28.4
2.995,11 = 2.60320
99% C.I.
df = 12 – 1 = 11
2.005,11 = 26.7569
(12  1)( 44.9)
(12  1)( 44.9)
< 2 <
2.60320
26.7569
18.46 < 2 < 189.73
b)
n=7
x = 4.37
s = 1.24
2.975,6 = 1.23734
s2 = 1.5376
95% C.I. df = 7 – 1 = 6
2.025,6 = 14.4494
(7  1)(1.5376)
(7  1)(1.5376)
< 2 <
1.23734
14.4494
0.64 < 2 < 7.46
c)
n = 20
x = 105
s = 32
2.95,19 = 10.11701
s2 = 1024
90% C.I.
df = 20 – 1 = 19
2.05,19 = 30.1435
(20  1)(1024)
(20  1)(1024)
< 2 <
10.11701
30.1435
645.45 < 2 < 1923.10
d)
n = 17
s2 = 18.56
2.90,16 = 9.31224
80% C.I.
df = 17 – 1 = 16
2.10,16 = 23.5418
(17  1)(18.56)
(17  1)(18.56)
< 2 <
9.31224
23.5418
12.61 < 2 < 31.89
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.36
n = 16
s2 = 37.1833
2.99,15 = 5.22936
df = 16 – 1 = 15
98% C.I.
2.01,15 = 30.5780
(16  1)(37.1833)
(16  1)(37.1833)
< 2 <
30.5780
5.22936
18.24 < 2 < 106.66
8.37
n = 20
s = 4.3
2.99,19 = 7.63270
s2 = 18.49
df = 20 – 1 = 19
98% C.I.
2.01,19 = 36.1908
(20  1)(18.49)
(20  1)(18.49)
< 2 <
36.1908
7.63270
9.71 < 2 < 46.03
Point Estimate = s2 = 18.49
8.38
n = 15
s2 = 3.067
2.995,14 = 4.07466
df = 15 – 1 = 14
99% C.I.
2.005,14 = 31.3194
(15  1)(3.067)
(15  1)(3.067)
< 2 <
31.3194
4.07466
1.37 < 2 < 10.54
8.39
n = 14
s2 = 26,798,241.76
95% C.I.
df = 14 – 1 = 13
Point Estimate = s2 = 26,798,241.76
2.975,13 = 5.00874
2.025,13 = 24.7356
(14  1)( 26,798,241.76)
(14  1)( 26,798,241.76)
< 2 <
24.7356
5.00874
14,084,038.51 < 2 < 69,553,848.45
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.40
a)
 = 36
n=
E=5
95% Confidence
z.025 = 1.96
z 2 2 (1.96) 2 (36) 2
= 199.15

E2
52
Sample 200
b)
 = 4.13
n=
E=1
99% Confidence
z 2 2 (2.575) 2 (4.13) 2

E2
12
z.005 = 2.575
= 113.1
Sample 114
c)
Range = 500 – 80 = 420
E = 10
1/4 Range = (.25)(420) = 105
90% Confidence
n =
z.05 = 1.645
z 2 2 (1.645) 2 (105) 2

= 298.3
E2
10 2
Sample 299
d)
Range = 108 – 50 = 58
E=3
1/4 Range = (.25)(58) = 14.5
88% Confidence
n =
z.06 = 1.555
z 2 2 (1.555) 2 (14.5) 2

E2
32
= 56.5
Sample 57
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.41
a)
E = .02
p = .40
96% Confidence
z.02 = 2.05
z 2 p  q (2.05) 2 (.40)(.60)
n =
= 2521.5

E2
(.02) 2
Sample 2522
b)
E = .04
p = .50
95% Confidence
z.025 = 1.96
z 2 p  q (1.96) 2 (.50)(.50)
= 600.25

E2
(.04) 2
n =
Sample 601
c)
E = .05
n =
p = .55
90% Confidence
z.05 = 1.645
z 2 p  q (1.645) 2 (.55)(.45)
= 267.9

E2
(.05) 2
Sample 268
d)
E =.01
n =
p = .50
99% Confidence
z.005 = 2.575
z 2 p  q (2.575) 2 (.50)(.50)

= 16,576.6
E2
(.01) 2
Sample 16,577
8.42
E = $200
 = $1,000
99% Confidence
z.005 = 2.575
z 2 2 (2.575) 2 (1000) 2

n =
= 165.77
E2
200 2
Sample 166
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.43
 = $12.50
E = $2.00
n =
90% Confidence
z.05 = 1.645
z 2 2 (1.645) 2 (12.50) 2
= 105.7

E2
2.00 2
Sample 106
8.44
E = $100
Range = $2,500 - $600 = $1,900
  1/4 Range = (.25)($1,900) = $475
90% Confidence
n =
z.05 = 1.645
z 2 2 (1.645) 2 (475) 2

= 61.05
E2
100 2
Sample 62
8.45
p = .20
q = .80
90% Confidence,
E = .02
z.05 = 1.645
z 2 p  q (1.645) 2 (.20)(.80)

n =
= 1082.41
E2
(.02) 2
Sample 1083
8.46
p = .50
q = .50
95% Confidence,
E = .05
z.025 = 1.96
z 2 p  q (1.96) 2 (.50)(.50)

n =
= 384.16
E2
(.05) 2
Sample 385
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.47 E = .10
p = .50
q = .50
95% Confidence,
z.025 = 1.96
z 2 p  q (1.96) 2 (.50)(.50)
n =
= 96.04

E2
(.10) 2
Sample 97
8.48 x = 45.6
 = 7.75
80% confidence
xz

n
n = 35
z.10 = 1.28
 45.6  1.28
7.75
35
= 45.6 + 1.68
43.92 <  < 47.28
94% confidence
xz

n
z.03 = 1.88
 45.6  1.88
7.75
35
= 45.6 + 2.46
43.14 <  < 48.06
98% confidence
xz

n
z.01 = 2.33
 45.6  2.33
7.75
35
= 45.6 + 3.05
42.55 <  < 48.65
x = 45.6 (point estimate)
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.49
x = 12.03 (point estimate)
/2 = .05
For 90% confidence:
xt
s
n
s = .4373
 12.03  1.833
(.4373)
10
n = 10
df = 9
t.05,9= 1.833
= 12.03 + .25
11.78 <  < 12.28
/2 = .025
For 95% confidence:
xt
s
n
 12.03  2.262
(.4373)
10
t.025,9 = 2.262
= 12.03 + .31
11.72 <  < 12.34
/2 = .005
For 99% confidence:
xt
s
n
 12.03  3.25
(.4373)
10
t.005,9 = 3.25
= 12.03 + .45
11.58 <  < 12.48
8.50
a)
n = 715
pˆ 
x = 329
95% confidence
z.025 = 1.96
329
= .46
715
pˆ  z
pˆ  qˆ
(.46)(.54)
= .46 + .0365
 .46  1.96
n
715
.4235 < p < .4965
b)
n = 284
pˆ  z
p̂ = .71
90% confidence
z.05 = 1.645
pˆ  qˆ
(.71)(.29)
 .71  1.645
= .71 + .0443
n
284
.6657 < p < .7543
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Chapter 8: Statistical Inference: Estimation for Single Populations
c)
n = 1250
pˆ  z
p̂ = .48
95% confidence
z.025 = 1.96
pˆ  qˆ
(.48)(.52)
= .48 + .0277
 .48  1.96
n
1250
.4523 < p < .5077
d)
n = 457
pˆ 
x = 270
98% confidence
z.01 = 2.33
270
= .591
457
pˆ  z
pˆ  qˆ
(.591)(.409)
= .591 + .0536
 .591  2.33
n
457
.5374 < p < .6446
8.51
n = 10
s2 = 54.7667
s = 7.40045
90% confidence,
2.95,9 = 3.32512
/2 = .05
df = 10 – 1 = 9
1 - /2 = .95
2.05,9 = 16.9190
(10  1)(54.7667)
(10  1)(54.7667)
< 2 <
16.9190
3.32512
29.133 < 2 < 148.235
95% confidence,
2.975,9 = 2.70039
/2 = .025
1 - /2 = .975
2.025,9 = 19.0228
(10  1)(54.7667)
(10  1)(54.7667)
< 2 <
19.0228
2.70039
25.911 < 2 < 182.529
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.52
a)
 = 44
n =
E=3
95% confidence
z.025 = 1.96
z 2 2 (1.96) 2 (44) 2
= 826.4

E2
32
Sample 827
b)
Range = 88 – 20 = 68
E=2
use  = 1/4(range) = (.25)(68) = 17
90% confidence
z.05 = 1.645
z 2 2 (1.645) 2 (17) 2

= 195.5
E2
22
Sample 196
c)
E = .04
p = .50
98% confidence
q = .50
z.01 = 2.33
z 2 p  q (2.33) 2 (.50)(.50)

= 848.3
E2
(.04) 2
Sample 849
d)
E = .03
p = .70
95% confidence
q = .30
z.025 = 1.96
z 2 p  q (1.96) 2 (.70)(.30)

= 896.4
E2
(.03) 2
Sample 897
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.53 n = 17
x = 10.765
/2 = .005
99% confidence
s
xt
n
 10.765  2.921
df = 17 – 1 = 16
s = 2.223
2.223
17
t.005,16 = 2.921
= 10.765 + 1.575
9.19 < µ < 12.34
8.54
p = .40
n =
E=.03
90% Confidence
z.05 = 1.645
z 2 p  q (1.645) 2 (.40)(.60)
= 721.61

E2
(.03) 2
Sample 722
8.55
s2 = 4.941
n = 17
2.995,16 = 5.14216
99% C.I.
df = 17 – 1 = 16
2.005,16 = 34.2671
(17  1)( 4.941)
(17  1)( 4.941)
< 2 <
34.2671
5.14216
2.307 < 2 < 15.374
8.56 n = 45
98% Confidence
xz
 = 48
x = 213

n
 213  2.33
z.01 = 2.33
48
45
= 213 ± 16.67
196.33 < µ < 229.67
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.57 n = 39
90% confidence
xz
 = 3.891
x = 37.256

n
z.05 = 1.645
 37.256  1.645
3.891
39
= 37.256 ± 1.025
36.231 < µ < 38.281
8.58
 = 6
E=1
98% Confidence
z.01 = 2.33
z 2 2 (2.33) 2 (6) 2
= 195.44

E2
12
n =
Sample 196
8.59
n = 1,255
pˆ 
x = 714
95% Confidence
z.025 = 1.96
714
= .569
1255
pˆ  z
pˆ  qˆ
(.569)(. 431)
 .569  1.96
= .569 ± .027
n
1,255
.542 < p < .596
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.60
n = 41
x  128
s = 21
98% C.I.
df = 41 – 1 = 40
t.01,40 = 2.423
Point Estimate = $128
s
xt
n
 128  2.423
21
41
= 128 + 7.947
120.053 <  < 135.947
Interval Width = 135.947 – 120.053 = 15.894
8.61
n = 60
x = 6.717
98% Confidence
xz

n
 = 3.06
N = 300
z.01 = 2.33
N n
3.06 300  60
=
 6.717  2.33
N 1
60 300  1
6.717 ± 0.825
5.892 < µ < 7.542
8.62 E = $20
Range = $600 – $30 = $570
1/4 Range = (.25)($570) = $142.50
95% Confidence
z.025 = 1.96
z 2 2 (1.96) 2 (142.50) 2

n =
= 195.02
E2
20 2
Sample 196
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.63
n = 245
pˆ 
x = 189
90% Confidence
z.05= 1.645
x 189

= .77
n 245
pˆ  qˆ
(.77)(.23)
= .77 ± .044
 .77  1.645
n
245
pˆ  z
.726 < p < .814
8.64
n = 90
pˆ 
x = 30
95% Confidence
z.025 = 1.96
x 30

= .33
n 90
pˆ  qˆ
(.33)(.67)
 .33  1.96
= .33 ± .097
n
90
pˆ  z
.233 < p < .427
8.65
n = 12
x = 43.7
s2 = 228
df = 12 – 1 = 11
95% C.I.
t.025,11 = 2.201
xt
s
n
 43.7  2.201
228
12
= 43.7 + 9.59
34.11 <  < 53.29
2.99,11 = 3.05350
2.01,11 = 24.7250
(12  1)( 228)
(12  1)( 228)
< 2 <
24.7250
3.05350
101.44 < 2 < 821.35
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.66
n = 27
x = 4.82
95% CI:
s
xt
n
s = 0.37
df = 26
t.025,26 = 2.056
0.37
 4.82  2.056
27
= 4.82 + .1464
4.6736 < µ < 4.9664
Since 4.50 is not in the interval, we are 95% confident that µ does not
equal 4.50.
8.67
n = 77
 = 12
x = 2.48
95% Confidence
xz

n
 2.48  1.96
z.025 = 1.96
12
77
= 2.48 ± 2.68
-0.20 < µ < 5.16
The point estimate is 2.48
The interval is inconclusive. It says that we are 95% confident that the average
arrival time is somewhere between .20 of a minute (12 seconds) early and 5.16
minutes late. Since zero is in the interval, there is a possibility that, on average,
the flights are on time.
8.68
p̂ =.33
n = 560
99% Confidence
pˆ  z
z.005= 2.575
pˆ  qˆ
(.33)(.67)
 .33  2.575
= .33 ± .05
n
560
.28 < p < .38
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.69
p = .50
E = .05
98% Confidence
z.01 = 2.33
z 2 p  q (2.33) 2 (.50)(.50)
= 542.89

E2
(.05) 2
Sample 543
8.70
n = 27
x = 2.10
/2 = .01
98% confidence
s
xt
n
 2.10  2.479
df = 27 – 1 = 26
s = 0.86
0.86
27
t.01,26 = 2.479
= 2.10 ± 0.41
1.69 < µ < 2.51
8.71
df = 23 – 1 = 22
n = 23
2.95,22 = 12.33801
s = .014419
90% C.I.
2.05,22 = 33.9245
(23  1)(.014419) 2
(23  1)(.014419) 2
< 2 <
33.9245
12.33801
.00013 < 2 < .00037
8.72
n = 39
xz
x = 1.294

n
 1.294  2.575
 = 0.205
0.205
39
99% Confidence
z.005 = 2.575
= 1.294 ± .085
1.209 < µ < 1.379
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Chapter 8: Statistical Inference: Estimation for Single Populations
8.73
The sample mean fill for the 510 cans is 354.373 ml with a standard deviation of
1.5857 ml. The 99% confidence interval for the population fill is 353.837 ml to
354.908 ml which does not include 355 ml. We are 99% confident that the
population mean is not 355 ml, indicating that the machine may be underfilling
the cans.
8.74
The point estimate for the average length of burn of the new bulb is 2198.217
hours. Eighty-four bulbs were included in this study. A 90% confidence interval
can be constructed from the information given. The error of the confidence
interval is + 27.76691. Combining this with the point estimate yields the 90%
confidence interval of 2198.217 + 27.76691 = 2170.450 < µ < 2225.984.
8.75
The point estimate for the average age of a first time buyer is 27.63 years. The
sample of 21 buyers produces a standard deviation of 6.54 years. We are 99%
confident that the actual population mean age of a first-time home buyer is
between 24.0222 years and 31.2378 years.
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