Chapter 8: Statistical Inference: Estimation for Single Populations 1
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.
Know the difference between point and interval estimation.
2.
Estimate a population mean from a sample mean when  is known.
3.
Estimate a population mean from a sample mean when  is unknown.
4.
Estimate a population proportion from a sample proportion.
5.
Estimate the population variance from a sample variance.
6.
Estimate the minimum sample size necessary to achieve given statistical
goals.
Chapter 8: Statistical Inference: Estimation for Single Populations 2
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.
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 favor. 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 to estimate
population means from small samples when  is unknown. Emphasize that this
applies only when the population is normally distributed. 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. In addition, note that
some business researchers always prefer to use the t distribution when  is
unknown.
Chapter 8: Statistical Inference: Estimation for Single Populations 3
A formula is given in chapter 8 for estimating the population variance.
Here 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, exercise extreme
caution is using this technique. Some statisticians omit this technique from
consideration.
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.
CHAPTER OUTLINE
8.1
Estimating the Population Mean Using the z Statistic.
Finite Correction Factor
Confidence Interval to Estimate µ When  is Unknown
Confidence Interval to Estimate  When the Population Standard
Deviation is Unknown and n is Large.
8.2
Estimating the Population Mean Using the t Statistic.
The t Distribution
Robustness
Characteristics of the t Distribution.
Reading the t Distribution Table
Confidence Intervals to Estimate µ When  is Unknown and
Sample Size is Small
Chapter 8: Statistical Inference: Estimation for Single Populations 4
8.3
Estimating the Population Proportion
8.4
Estimating the Population Variance
8.5
Estimating Sample Size
Sample Size When Estimating µ
Determining Sample Size When Estimating p
KEY WORDS
Bounds
Chi-square Distribution
Degrees of Freedom(df)
Error of Estimation
Interval Estimate
Point Estimate
Robust
Sample-Size Estimation
t Distribution
t Value
Chapter 8: Statistical Inference: Estimation for Single Populations 5
SOLUTIONS TO PROBLEMS IN CHAPTER 8
8.1 a)
x = 25  = 3.5
n = 60
95% Confidence
x + z

n
z.025 = 1.96
= 25 + 1.96
3 .5
= 25 + 0.89 = 24.11 < µ < 25.89
60
b) x = 119.6
s = 23.89
98% Confidence
s
x + z
n
c) x = 3.419
90% C.I.
x + z
s
n
d) x = 56.7
80% C.I.
x ±z

n
= 119.6 + 2.33
n = 75
z.01 = 2.33
2.89
= 119.6 ± 6.43 = 113.17 < µ < 126.03
75
s = 0.974
z.05 = 1.645
= 3.419 + 1.645
n = 32
0.974
= 3.419 ± .283 = 3.136 < µ < 3.702
32
 = 12.1
N = 500
n = 47
z.10 = 1.28
N n
12.1 500  47
= 56.7 + 1.28
N 1
47 500  1
56.7 ± 2.15 = 54.55 < µ < 58.85
=
Chapter 8: Statistical Inference: Estimation for Single Populations 6
8.2 n = 36
95% C.I.
x ± z

n
8.3 n = 81
90% C.I.
x ± z
s
n
x = 90.4
94% C.I.

n
8.5 n = 39
96% C.I.
x ± z
2
=
36
= 211 ± 1.96
x = 47
s
n
= 47 ± 1.645
5.89
= 47 ± 1.08 = 45.92 < µ < 48.08
81
x = 90.4
Point Estimate
z.03 = 1.88
= 90.4 ± 1.88
N = 200
z.02 = 2.05
49
= 90.4 ± 1.57 = 88.83 < µ < 91.97
70
x = 66
N n
11
= 66 ± 2.05
N 1
9
66 ± 3.25 = 62.75 < µ < 69.25
x = 66
211 ± 7.51 = 203.49 < µ < 218.51
s = 5.89
z.05=1.645
2 = 49
8.4 n = 70
x + z
 = 23
x = 211
z.025 = 1.96
Point Estimate
s = 11
200  39
=
200  1
Chapter 8: Statistical Inference: Estimation for Single Populations 7
8.6 n = 120
99% C.I.
x = 18.72
z.005 = 2.575
s = 0.8735
x = 18.72 Point Estimate
x + z
s
n
= 18.72 ± 2.575
8.7 N = 1500
95% C.I.
n = 187
x = 5.3 years
z.025 = 1.96
x = 5.3 years
x ± z
s
n
0.8735
= 8.72 ± .21 = 18.51 < µ < 18.93
120
s = 1.28 years
Point Estimate
1.28 1500  187
N n
= 5.3 ± 1.96
=
N 1
1500  1
187
5.3 ± .17 = 5.13 < µ < 5.47
8.8 n = 32
90% C.I.
x ± z
s
n
8.9 n = 36
98% C.I.
x ± z
s
n
x = 5.656
z.05 = 1.645
s = 3.229
= 5.656 ± 1.645
x = 3.306
z.01 = 2.33
= 3.306 ± 2.33
3.229
= 5.656 ± .939 = 4.717 < µ < 6.595
32
s = 1.167
1.167
= 3.306 ± .453 = 2.853 < µ < 3.759
36
Chapter 8: Statistical Inference: Estimation for Single Populations 8
8.10 n = 36
x = 2.139
s = .113
x = 2.139 Point Estimate
90% C.I.
x ± z
s
n
8.11  = 27.4
x = 24.533
z.05 = 1.645
= 2.139 ± 1.645
(.113)
= 2.139 ± .03 = 2.109 < µ < 2.169
36
95% confidence interval
n = 45
s = 5.1239
z = + 1.96
Confidence interval:
x + z
s
n
= 24.533 + 1.96
24.533 + 1.497 = 23.036 <  < 26.030
8.12 The point estimate is 0.5765.
n = 41
The assumed standard deviation is 0.1394
99% level of confidence: z = + 1.96
Confidence interval:
0.5336 < µ < 0.6193
Error of the estimate:
0.6193 - 0.5765 = 0.0428
5.1239
=
45
Chapter 8: Statistical Inference: Estimation for Single Populations 9
8.13 n = 13
x = 45.62
s = 5.694
df = 13 – 1 = 12
95% Confidence Interval
/2=.025
t.025,12 = 2.179
xt
s
n
= 45.62 ± 2.179
8.14 n = 12
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
xt
s
n
t.05,11 = 1.796
= 319.17 ± (1.796)
8.15 n = 27
x = 128.4
9.104
= 319.17 ± 4.72 = 314.45 < µ < 323.89
12
s = 20.64
df = 27 – 1 = 26
98% Confidence Interval
/2=.01
t.01,26 = 2.479
xt
s
n
= 128.4 ± 2.479
20.6
= 128.4 ± 9.83 = 118.57 < µ < 138.23
27
x = 128.4 Point Estimate
Chapter 8: Statistical Inference: Estimation for Single Populations 10
8.16 n = 15
x = 2.364
df = 15 – 1 = 14
s2 = 0.81
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
= 16.088 ± .457 = 15.631 < µ < 16.545
25
x = 16.088 Point Estimate
8.18 n = 22
x = 1,192
98% CI and /2 = .01
xt
s
n
= 1,192 + (2.518)
s = 279
df = n - 1 = 21
t.01,21 = 2.518
279
= 1,192 + 149.78 = 1,042.22 <  < 1,341.78
22
Chapter 8: Statistical Inference: Estimation for Single Populations 11
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 = 5.335
s = 2.016
df = 28 – 1 = 27
/2=.05
90% Confidence Interval
t.05,27 = 1.703
xt
s
n
= 5.335 ± 1.703
8.21 n = 10
95% Confidence
xt
s
n
2.016
= 5.335 + .649 = 4.686 < µ < 5.984
28
x = 49.8
s = 18.22
/2=.025
t.025,9 = 2.262
= 49.8 ± 2.262
df = 10 – 1 = 9
18.22
= 49.8 + 13.03 = 36.77 < µ < 62.83
10
Chapter 8: Statistical Inference: Estimation for Single Populations 12
8.22 n = 14,
/2 = .01,
98% confidence,
df = 13
t.01,13 = 2.650
from data:
x = 152.16
s = 14.42
xt
confidence interval:
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 a) n = 44
pˆ  z
p̂ =.51
pˆ  qˆ
n
b) n = 300
pˆ  z
d) n = 95
pˆ  z
(.51)(.49)
= .51 ± .194 = .316 < p< .704
44
= .51 ± 2.575
p̂ = .82
z.005 = 2.575
95% C.I.
z.025 = 1.96
(.82)(.18)
pˆ  qˆ
= .82 ± 1.96
= .82 ± .043 = .777 < p < .863
n
300
c) n = 1150
pˆ  z
99% C.I.
p̂ = .48
90% C.I.
z.05 = 1.645
(.48)(.52)
pˆ  qˆ
= .48 ± 1.645
= .48 ± .024 = .456 < p < .504
n
1150
p̂ = .32
88% C.I.
z.06 = 1.555
(.32)(.68)
pˆ  qˆ
= .32 ± 1.555
= .32 ± .074 = .246 < p < .394
n
95
Chapter 8: Statistical Inference: Estimation for Single Populations 13
8.24 a) n = 116
p̂ =
pˆ  qˆ
n
c) n = 240
97% C.I.
z.015 = 2.17
= .60 ± 2.17
x = 106
(.60)(.40)
800
= .60 ± .038 = .562 < p < .638
85% C.I.
z.075 = 1.44
x 106

= .44
n 240
(.44)(.56)
pˆ  qˆ
= .44 ± 1.44
= .44 ± .046 = .394 < p < .486
n
240
pˆ  z
d) n = 60
pˆ  z
x = 479
x 479

= .60
n 800
pˆ  z
p̂ =
z.005 = 2.575
(.49)(.51)
pˆ  qˆ
= .49 ± 2.575
= .49 ± .12 = .37 < p < .61
n
116
b) n = 800
p̂ =
99% C.I.
x
57

= .49
n 116
pˆ  z
p̂ =
x = 57
x = 21
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
Chapter 8: Statistical Inference: Estimation for Single Populations 14
8.25
n = 85
p̂ =
pˆ  z
x = 40
(.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 ± .106 = .364 < p < .576
n
85
99% C.I.
pˆ  z
z.05 = 1.645
x 40

= .47
n 85
95% C.I.
pˆ  z
90% C.I.
z.005 = 2.575
(.47)(.53)
pˆ  qˆ
= .47 ± 2.575
= .47 ± .14 = .33 < p < .61
n
85
All things being constant, as the confidence increased, the width of the interval
increased.
8.26
n = 1003
pˆ  z
p̂ = .245
99% CI
z.005 = 2.575
(.245)(.755)
pˆ  qˆ
= .245 + 2.575
= .245 + .035 = .21 < p < .28
n
1003
Chapter 8: Statistical Inference: Estimation for Single Populations 15
8.27
n = 560
p̂ = .47
n = 560
8.28
pˆ  z
8.29
b)
90% CI
z.05 = 1.645
(.28)(.72)
pˆ  qˆ
= .28 + 1.645
= .28 + .0312 = .2488 < p < .3112
n
560
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
n = 3481
p̂ =
a)
p̂ = .28
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 = 927
x
927

= .266
n 3481
p̂ = .266 Point Estimate
99% C.I.
pˆ  z
z.005 = 2.575
(.266)(.734)
pˆ  qˆ
= .266 + 2.575
= .266 ± .02 =
n
3481
.246 < p < .286
Chapter 8: Statistical Inference: Estimation for Single Populations 16
8.30
n = 89
p̂ =
85% C.I.
z.075 = 1.44
x 48

= .54
n 89
pˆ  z
8.31
x = 48
(.54)(.46)
pˆ  qˆ
= .54 ± 1.44
= .54 ± .076 = .464 < p < .616
n
89
p̂ = .63
pˆ  z
8.32 a) n = 12
n = 672
95% Confidence
z = + 1.96
(.63)(.37)
pˆ  qˆ
= .63 + 1.96
= .63 + .0365 = .5935 < p < .6665
n
672
x = 28.4
2.995,11 = 2.60321
s2 = 44.9
99% C.I.
2.005,11 = 26.7569
(12  1)( 44.9)
(12  1)( 44.9)
< 2 <
26.7569
2.60321
18.46 < 2 < 189.73
b) n = 7
x = 4.37
95% C.I.
2.975,6 = 1.237347
s = 1.24
s2 = 1.5376
df = 12 – 1 = 11
2.025,6 = 14.4494
(7  1)(1.5376)
(7  1)(1.5376)
< 2 <
14.4494
1.237347
0.64 < 2 < 7.46
c) n = 20
x = 105
90% C.I.
s = 32
s2 = 1024
df = 20 – 1 = 19
df = 12 – 1 = 11
Chapter 8: Statistical Inference: Estimation for Single Populations 17
2.95,19 = 10.117
2.05,19 = 30.1435
(20  1)(1024)
(20  1)(1024)
< 2 <
30.1435
10.117
645.45 < 2 < 1923.10
d) n = 17
s2 = 18.56
2.90,16 = 9.31223
80% C.I.
df = 17 – 1 = 16
2.10,16 = 23.5418
(17  1)(18.56)
(17  1)(18.56)
< 2 <
23.5418
9.31223
12.61 < 2 < 31.89
8.33
n = 16
s2 = 37.1833
2.99,15 = 5.22935
98% C.I.
df = 16-1 = 15
2.01,15 = 30.5779
(16  1)(37.1833)
(16  1)(37.1833)
< 2 <
30.5779
5.22935
18.24 < 2 < 106.66
8.34
n = 12
s = 4.3
s2 = 18.49
98% C.I.
df = 20 – 1 = 19
2.99,19 = 7.63273
2.01,19 = 36.1980
(20  1)(18.49)
(20  1)(18.49)
< 2 <
36.1980
7.63273
9.71 < 2 < 46.03
Point Estimate = s2 = 18.49
8.35
n = 152
s2 = 3.067
2.995,14 = 4.07468
99% C.I.
2.005,14 = 31.3193
df = 15 – 1 = 14
Chapter 8: Statistical Inference: Estimation for Single Populations 18
(15  1)(3.067)
(15  1)(3.067)
< 2 <
31.3193
24.07468
1.37 < 2 < 10.54
8.36
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
8.37 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
E=1
99% Confidence
z 2 2 (2.575) 2 (4.13) 2

n=
E2
12
= 113.1
Sample 114
c) E = 10
Range = 500 - 80 = 420
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
z.005 = 2.575
Chapter 8: Statistical Inference: Estimation for Single Populations 19
d) E = 3
Range = 108 - 50 = 58
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
8.38 a) E = .02
n =
p=.40
96% Confidence
z.02 = 2.05
z 2 p  q (2.05) 2 (.40)(.60)
= 2521.5

E2
(.02) 2
Sample 2522
b) E = .04
n =
p=.50
95% Confidence
z.025 = 1.96
z 2 p  q (1.96) 2 (.50)(.50)

= 600.25
E2
(.04) 2
Sample 601
c) E = .05
n =
p = .55
90% Confidence
z 2 p  q (1.645) 2 (.55)(.45)

= 267.9
E2
(.05) 2
Sample 268
z.05 = 1.645
Chapter 8: Statistical Inference: Estimation for Single Populations 20
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
8.39
E = $200
n =
 = $1,000
99% Confidence
z.005 = 2.575
z 2 2 (2.575) 2 (1000) 2
= 165.77

E2
200 2
Sample 166
8.40
E = $2
n =
 = $12.50
90% Confidence
z 2 2 (1.645) 2 (12.50) 2

= 105.7
E2
22
Sample 106
8.41
E = $100
Range = $2,500 - $600 = $1,900
  1/4 Range = (.25)($1,900) = $475
90% Confidence
z.05 = 1.645
z 2 2 (1.645) 2 (475) 2

n =
= 61.05
E2
100 2
Sample 62
z.05 = 1.645
Chapter 8: Statistical Inference: Estimation for Single Populations 21
8.42 p = .20
q = .80
90% Confidence,
n =
E = .02
z.05 = 1.645
z 2 p  q (1.645) 2 (.20)(.80)
= 1082.41

E2
(.02) 2
Sample 1083
8.43 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
8.44 E = .10
p = .50
95% Confidence,
q = .50
z.025 = 1.96
z 2 p  q (1.96) 2 (.50)(.50)

n =
= 96.04
E2
(.10) 2
Sample 97
Chapter 8: Statistical Inference: Estimation for Single Populations 22
8.45 x = 45.6
s = 7.7467
80% confidence
xz
s
n
n = 35
z.10 = 1.28
 45.6  1.28
7.7467
35
= 45.6 + 1.676
43.924 <  < 47.276
94% confidence
z.03 = 1.88
s
7.7467
xz
n
 45.6  1.88
35
= 45.6 + 2.462
43.138 <  < 48.062
98% confidence
z.01 = 2.33
s
7.7467
xz
n
 45.6  2.33
35
= 45.6 + 3.051
42.549 <  < 48.651
8.46
x = 12.03 (point estimate)
For 90% confidence:
xt
s
n
 12.03  1.833
11.78 <  < 12.28
s = .4373
/2 = .05
.4373
10
n = 10
t.05,9= 1.833
= 12.03 + .25
df = 9
Chapter 8: Statistical Inference: Estimation for Single Populations 23
/2 = .025
For 95% confidence:
s
xt
n
 12.03  2.262
t.025,9 = 2.262
.4373
= 12.03 + .31
10
11.72 <  < 12.34
/2 = .005
For 99% confidence:
s
xt
n
 12.03  3.25
t.005,9 = 3.25
(.4373)
= 12.03 + .45
10
11.58 <  < 12.48
8.47 a) n = 715
pˆ 
x = 329
329
= .46
715
95% confidence
pˆ  z
z.025 = 1.96
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
Chapter 8: Statistical Inference: Estimation for Single Populations 24
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
8.48
n = 10
s2 = 54.7667
s = 7.40045
90% confidence,
/2 = .05
2.95,9 = 3.32511
df = 10 – 1 = 9
1 - /2 = .95
2.05,9 = 16.919
(10  1)(54.7667)
(10  1)(54.7667)
< 2 <
16.919
3.32511
29.133 < 2 < 148.236
95% confidence,
/2 = .025
2.975,9 = 2.70039
1 - /2 = .975
2.025,9 = 19.0228
(10  1)(54.7667)
(10  1)(54.7667)
< 2 <
19.0228
2.70039
4.258 < 2 < 182.529
Chapter 8: Statistical Inference: Estimation for Single Populations 25
8.49 a)  = 44
n =
E=3
95% confidence
z 2 2 (1.96) 2 (44) 2
= 826.4

E2
32
Sample 827
b) E = 2
Range = 88 - 20 = 68
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 4(1.96) 2 (.70)(.30)

= 896.4
E2
(.03) 2
Sample 897
z.025 = 1.96
Chapter 8: Statistical Inference: Estimation for Single Populations 26
8.50
n = 17
x = 10.765
/2 = .005
99% confidence
s
xt
n
S = 2.223
 10.765  2.921
df = 17 - 1 = 16
t.005,16 = 2.921
2.223
= 10.765 + 1.575
17
9.19 < µ < 12.34
8.51
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
8.52
s2 = 4.941
n = 17
2.995,16 = 5.14224
99% C.I.
df = 17 – 1 = 16
2.005,16 = 34.2672
(17  1)( 4.941)
(17  1)( 4.941)
< 2 <
34.2672
5.14224
2.307 < 2 < 15.374
8.53 n = 45
x = 213
98% Confidence
xz
s
n
 213  2.33
196.33 < µ < 229.67
s = 48
z.01 = 2.33
48
45
= 213 ± 16.67
Chapter 8: Statistical Inference: Estimation for Single Populations 27
8.54 n = 39
x = 37.256
s = 3.891
90% confidence
z.05 = 1.645
s
3.891
xz
n
 37.256  1.645
39
= 37.256 ± 1.025
36.231 < µ < 38.281
8.55
=6
E=1
98% Confidence
z.98 = 2.33
z 2 2 (2.33) 2 (6) 2
= 195.44

E2
12
n =
Sample 196
8.56
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
Chapter 8: Statistical Inference: Estimation for Single Populations 28
8.57 n = 25
s = 21
x = 128
df = 25 – 1 = 24
98% C.I.
t.01,24 = 2.492
Point Estimate = $128
s
xt
n
 128  2.492
21
25
= 128 + 10.466
117.534 <  < 138.466
Interval Width = 138.466 – 117.534 = 20.932
8.58
n = 60
x = 6.717
98% Confidence
xz
s
n
s = 3.059
N =300
z.01 = 2.33
N n
3.059 300  60
 6.717  2.33
N 1
300  1
60
6.717 ± 0.824 =
5.893 < µ < 7.541
8.59 E = $20
Range = $600 - $30 = $570
1/4 Range = (.25)($570) = $142.50
95% Confidence
n =
z.025 = 1.96
z 2 2 (1.96) 2 (142.50) 2

= 195.02
E2
20 2
Sample 196
=
Chapter 8: Statistical Inference: Estimation for Single Populations 29
8.60
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
8.61
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
8.62
n = 12
x = 43.7
s2 = 228
df = 12 – 1 = 11
t.025,11 = 2.201
xt
s
n
 43.7  2.201
228
12
= 43.7 + 9.59
34.11 <  < 53.29
2.975,11 = 3.81575
2.025,11 = 21.92
(12  1)( 228)
(12  1)( 228)
< 2 <
21.92
3.81575
114.42 < 2 < 657.28
95% C.I.
Chapter 8: Statistical Inference: Estimation for Single Populations 30
8.63
n = 27
x = 4.82
95% CI:
xt
s
n
s = 0.37
df = 26
t.025,26 = 2.056
 4.82  2.056
0.37
27
= 4.82 + .1464
4.6736 < µ < 4.9664
We are 95% confident that µ does not equal 4.50.
8.64
n = 77
x = 2.48
s = 12
95% Confidence
xz
s
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.
8.65
p̂ =.33
n = 560
99% Confidence
pˆ  z
z.005= 2.575
pˆ  qˆ
(.33)(.67)
 .33  2.575
= .33 ± (2.575) = .33 ± .05
n
560
.28 < p < .38
Chapter 8: Statistical Inference: Estimation for Single Populations 31
8.66
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
8.67
n = 27
x = 2.10
98% confidence
xt
s
n
s = 0.86
/2 = .01
 2.10  2.479
0.86
27
df = 27 - 1 = 26
t.01,26 = 2.479
= 2.10 ± (2.479) = 2.10 ± 0.41
1.69 < µ < 2.51
8.68
df = 23 – 1 = 22
n = 23
2.95,22 = 12.338
s = .0631455
90% C.I.
2.05,22 = 33.9244
(23  1)(. 0631455) 2
(23  1)(. 0631455) 2
< 2 <
33.9244
12.338
.0026 < 2 < .0071
8.69
n = 39
xz
x = 1.294
s
n
 1.294  2.575
1.209 < µ < 1.379
s = 0.205
0.205
39
99% Confidence
z.005 = 2.575
= 1.294 ± (2.575) = 1.294 ± .085
Chapter 8: Statistical Inference: Estimation for Single Populations 32
8.70
The sample mean fill for the 58 cans is 11.9788 oz. with a standard deviation of
.0556 oz. The 99% confidence interval for the population fill is 11.9607 oz. to
11.9970 oz. which does not include 12 oz. We are 99% confident that the
population mean is not 12 oz. indicating an underfill from the machine.
8.71
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.
8.72
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 98%
confident that the actual population mean age of a first-time home buyer is
between 24.02 years and 31.24 years.
8.73
A poll of 781 American workers was taken. Of these, 506 drive their cars to
work. Thus, the point estimate for the population proportion is 506/781 = .648. A
95% confidence interval to estimate the population proportion shows that we are
95% confident that the actual value lies between .613 and .681. The error of this
interval is + .034.