Chapter 8: Statistical Inference: Estimation for Single Populations 1
Chapter 8
Estimating Parameters 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.
Estimate the population mean with a known population standard
deviation with the z statistic, correcting for a finite population if
necessary.
Estimate the population mean with an unknown population standard
deviation using the t statistic and properties of the t distribution.
Estimate a population proportion using the z statistic.
Use the chi-square distribution to estimate the population variance given
the sample variance.
Determine the sample size needed in order to estimate the population
mean and population proportion.
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
Chapter 8: Statistical Inference: Estimation for Single Populations 2
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 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.
Chapter 8: Statistical Inference: Estimation for Single Populations 3
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
Using the Computer to Construct Confidence Intervals of 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
Chapter 8: Statistical Inference: Estimation for Single Populations 4
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
SOLUTIONS TO PROBLEMS IN CHAPTER 8
 =3
n = 49
x = 20
95% Confidence
z.025 = 1.96
3

= 20 + 1.96
= 20 + 0.84 = 19.16 ≤ µ ≤ 20.84
xz
49
n
b) x = 115
 = 25
n = 81
98% Confidence
z.01 = 2.33
25

xz
= 115 + 2.33
= 115 ± 6.47 = 108.53 ≤ µ ≤ 121.47
81
n
c) x = 3.419
 = 1.974
n = 36
90% C.I.
z.05 = 1.645
1.974

= 3.419 + 1.645
= 3.419 ± .541 = 2.88 ≤ µ ≤ 3.96
xz
36
n
8.1 a)
d)
x = 66.7
80% C.I.
xz

n
 = 12.9
z.10 = 1.28
N = 500
N n
12.9 500  54
= 66.7 + 1.28
N 1
54 500  1
66.7 ± 2.12 = 64.58 ≤ µ ≤ 68.82
n = 54
=
Chapter 8: Statistical Inference: Estimation for Single Populations 5
8.2 n = 36
95% C.I.
xz

n
 = 23
x = 211
z.025 = 1.96
= 211 ± 1.96
23
36
=
211 ± 7.51 = 203.49 < µ < 218.51
 = 4.49
x = 49
z.05=1.645

4.49
= 49 ± 1.645
= 49 ± 0.74 = 48.26 ≤ µ ≤ 49.74
xz
n
100
8.3 n = 100
90% C.I.
2 = 49
8.4 n = 70
x = 90.4
94% C.I.
xz

n
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
 = 10
x = 56
z.02 = 2.05
 N n
10 200  49
xz
= 56 ± 2.05
=
n N 1
49 200  1
56 ± 2.55 = 53.45 ≤ µ ≤ 58.55
x = 56 Point Estimate
8.5 n = 49
96% C.I.
8.6 n = 120
99% C.I.
 = 0.8735
x = 18.72
z.005 = 2.575
x = 18.72 Point Estimate
xz

n
8.7 N = 1500
95% C.I.
= 18.72 ± 2.575
n = 190
z.025 = 1.96
0.8735
= 8.72 ± .21 = 18.51 < µ < 18.93
120
x = 5.5 years
= 1.25 years
Chapter 8: Statistical Inference: Estimation for Single Populations 6
x = 5.5 years Point Estimate
1.25 1500  190
 N n
= 5.5 ± 1.96
=
xz
190 1500  1
n N 1
5.5 ± .17 = 5.67 ≤ µ ≤ 5.33
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
8.11 95% confidence interval

x = 24.74
xz

= 24.74 + 1.96
n = 50
z = + 1.96
5.4
=
50
n
24.74 + 1.497 = 23.243 ≤  ≤ 26.237
Chapter 8: Statistical Inference: Estimation for Single Populations 7
8.12 The point estimate is 0.96 or 96 cents per pound.
n = 41
The assumed standard deviation is 0.14
95% level of confidence: z = + 1.96
Confidence interval:
0.917147 < µ < 1.002853
Error of the estimate:
1.002853 - 0.96 = 0.042853
8.13 n = 16
df = 16 – 1 = 15
x = 46 s = 5.22
95% Confidence Interval and /2=.025
t.025,15 = 2.131
s
5.22
xt
= 46 ± 2.131
= 46 ± 3.44 = 42.56 ≤ µ ≤ 49.44
n
16
8.14 n = 12
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 = 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
x = 2.364
s2 = 0.81
df = 15 – 1 = 14
Chapter 8: Statistical Inference: Estimation for Single Populations 8
90% Confidence interval
/2 = .05
t.05,14 = 1.761
xt
s
n
= 2.364 ± 1.761
8.17 n = 25
x = 26.088
99% Confidence Interval
/2 = .005
0.81
= 2.364 ± .409 = 1.955 < µ < 2.773
15
s = .817
df = 25 – 1 = 24
t.005,24 = 2.797
s
(.817)
xt
= 26.088 ± 2.797
= 26.088 ± .457 = 25.631 ≤ µ ≤ 26.545
n
25
x = 26.088 Point Estimate
Chapter 8: Statistical Inference: Estimation for Single Populations 9
8.18 n = 51
x = 1,192
98% CI and /2 = .01
xt
s
n
= 1,192 + 2.403
s = 279
df = n - 1 = 50
t.01,50 = 2.403
279
= 1,192 + 93.88 = 1,098.12 <  < 1,285.88
51
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 = 18
df = 17
95% CI
t.025,17 = 2.11
s = 0.1717
x = 2.5006
0.1717
2.5006 + 2.11
= 2.5006 + 0.0854 = 2.4152 ≤  ≤ 2.5860
18
Point Estimate = 2.5006
Error = 0.0006
8.20 n = 28
x = 5.335
90% Confidence Interval
s = 2.016
df = 28 – 1 = 27
/2 = .05
t.05,27 = 1.703
xt
8.21
s
n
= 5.335 ± 1.703
2.016
= 5.335 + .649 = 4.686 < µ < 5.984
28
n = 10
s = 15.903
df = 10 – 1 = 9
x = 55.3
95% Confidence
/2 = .025
t.025,9 = 2.262
s
15 .903
xt
= 55.3 ± 2.262
= 55.3+ 11.38 = 43.92 ≤ µ ≤ 66.68
n
10
8.22 n = 14
98% confidence
/2 = .01
t.01,13 = 2.650
from data:
x = 389.88
s = 105.30
df = 13
Chapter 8: Statistical Inference: Estimation for Single Populations 10
confidence interval:
xt
s
n
= 389.88 + 2.65
105.30 =
14
389.88 + 74.58 = 315.30 <  < 464.46
The point estimate is 389.88
/2 = .005
8.23
n = 40
df = 40 – 1 = 39
99% confidence
t.005,39 = 2.708
from data:
s = 8.21
x = 11.8
s
8.21
confidence interval: x  t
= 11.8 + 2.708
=
n
40
11.8 + 3.52 = 8.28 ≤  ≤ 15.32
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 26.011.
8.25
a)
n = 50
pˆ  z
b)
d)
p̂ = .85
n = 105
pˆ  z
z.025 = 1.96
99% C.I.
z.005 = 2.575
(.85)(.15)
pˆ  qˆ
= .85 ± 2.575
= .85 ± .046 = .804 ≤ p ≤ .896
400
n
n = 1100
pˆ  z
95% C.I.
(.55)(.45)
pˆ  qˆ
= .55 ± 1.96
= .55 ± .138 = .412 ≤ p ≤ .688
50
n
n = 400
pˆ  z
c)
p̂ =.55
p̂ = .58
90% C.I.
z.05 = 1.645
(.58)(.42)
pˆ  qˆ
= .58 ± 1.645
= .58 ± .025 = .555 ≤ p ≤ .605
1100
n
p̂ = .33
88% C.I.
z.06 = 1.555
(.33)(.67)
pˆ  qˆ
= .33 ± 1.555
= .33 ± .071 = .259 ≤ p ≤ .401
105
n
Chapter 8: Statistical Inference: Estimation for Single Populations 11
8.26
a)
n = 116
p̂ =
pˆ  z
8.27
n = 75
x = 106
85% C.I.
z.075 = 1.44
(.44)(.56)
pˆ  qˆ
= .44 ± 1.44
= .44 ± .046 = .394 < p < .486
n
240
n = 60
p̂ =
z.015 = 2.17
x 106
= .44

n 240
pˆ  z
d)
97% C.I.
(.60)(.40)
pˆ  qˆ
= .60 ± 2.17
= .60 ± .038 = .562 < p < .638
n
800
n = 240
p̂ =
x = 479
x 479
= .60

n 800
pˆ  z
c)
z.005 = 2.575
(.49)(.51)
pˆ  qˆ
= .49 ± 2.575
= .49 ± .12 = .37 < p < .61
116
n
n = 800
p̂ =
99% C.I.
x 57

= .49
n 116
pˆ  z
b)
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
60
n
x = 35
90% C.I.
z.05 = 1.645
x 35
= .47
p̂ =

n 75
(.47)(.53)
pˆ  qˆ
= .47 ± 1.645
= .47 ± .09 = .38 ≤ p ≤ .56
pˆ  z
85
n
95% C.I.
z.025 = 1.96
Chapter 8: Statistical Inference: Estimation for Single Populations 12
(.47)(.53)
pˆ  qˆ
= .47 ± 1.96
= .47 ± .11 = .36 ≤ p ≤ .58
85
n
pˆ  z
99% C.I.
z.005 = 2.575
(.47)(.53)
pˆ  qˆ
= .47 ± 2.575
= .47 ± .14 = .33 ≤ p ≤ .61
pˆ  z
85
n
All other things being constant, as the confidence increased, the width of the
interval increased.
8.28 a) n = 1003
pˆ  z
p̂ = .255
z.005 = 2.575
pˆ  qˆ
(.255)(.745)
= .255 + 2.575
= .255 + .035 = .220 < p < .290
n
1003
b) n = 10,000
pˆ  z
99% CI
p̂ = .255
99% CI
z.005 = 2.575
(.255)(.745)
pˆ  qˆ
= .255 + 2.575
= .255 + .011 = .244 < p < .266
10,000
n
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
(a) n = 660
p̂ = .47
95% CI
z.025 = 1.96
(.47)(.53)
pˆ  qˆ
= .47 + 1.96
= .47 + .0381 = .4319 ≤ p ≤ .5081
660
n
(b) n = 660
99% CI
z.005 = 2.575
p̂ = .28
pˆ  z
pˆ  z
8.30
n = 1250
p̂ =
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 = 3500
x = 947
pˆ  z
8.31
(.28)(.72)
pˆ  qˆ
= .28 + 2.575
= .28 + .0450 = .235 ≤ p ≤ .325
660
n
Chapter 8: Statistical Inference: Estimation for Single Populations 13
a)
x 947

= .271
n 3500
p̂ = .271 Point Estimate
b)
99% C.I.
p̂ =
pˆ  z
z.005 = 2.575
pˆ  qˆ
(.271)(.729)
= .271 + 2.575
= .271 ± .019 =
n
3500
.252 ≤ p ≤ .290
8.32
n = 89
p̂ =
x = 48
(.54)(.46)
pˆ  qˆ
= .54 ± 1.44
= .54 ± .076 = .464 < p < .616
n
89
p̂ = .63
n = 275
p̂ =
pˆ  z
8.35
a)
n = 690
95% Confidence
z.025 = + 1.96
(.63)(.37)
pˆ  qˆ
= .63 + 1.96
= .63 + .0360 = .594 ≤ p ≤ .666
690
n
pˆ  z
8.34
z.075 = 1.44
x 48

= .54
n 89
pˆ  z
8.33
85% C.I.
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
n = 15
x = 28.4
2.995,14 = 4.07466
s2 = 34.9 99% C.I.
 .005,14 = 31.3194
2
df = 15 – 1 = 14
Chapter 8: Statistical Inference: Estimation for Single Populations 14
b)
8.36
(15  1)(34.9)
(15  1)(34.9)
< 2 <
31.3194
4.07466
15.60 ≤ 2 ≤ 119.91
n = 7 x = 5.37 s = 2.24 s2 = 5.0176
2.975,6 = 1.23734
2.025,6 = 14.4494
(7  1)(5.0176)
(7  1)(5.0176)
≤ 2 ≤
14.4494
1.23734
2
2.08 ≤  ≤ 24.33
95% C.I. df = 7 – 1 = 6
c)
n = 19
x = 110 s = 32 s2 = 1024 90% C.I.
2.95,18 = 9.39045
2.05,18 = 28.8693
(19  1)(1024)
(19  1)(1024)
≤ 2 ≤
28.8693
9.39045
283.76 ≤ 2 ≤ 872.38
d)
n = 16
s2 = 19.56
80% C.I.
2
2
 .90,15 = 8.54675
 .10,15 = 22.3071
(16  1)(19.56)
(
16  1)(19.56)
≤ 2 ≤
22.3071
8.54675
4.38 ≤ 2 ≤ 34.33
n = 16
s2 = 37.1833
2.99,15 = 5.22936
98% C.I.
df = 19 – 1 = 18
df = 16 – 1 = 15
df = 16-1 = 15
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 = 18
s = 4.5
s2 = 20.25
98% C.I.
2
2
 .99,17 = 6.40774
 .01,17 = 33.4087
(18  1)(20.25)
(18  1)(20.25)
≤ 2 ≤
33.4087
6.40774
2
10.30 ≤  ≤ 53.72
Point Estimate = s2 = 20.25
df = 18 – 1 = 17
Chapter 8: Statistical Inference: Estimation for Single Populations 15
8.38
s2 = 3.067
n = 15
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
s2 = 26,080,949.45
n = 14
95% C.I.
df = 14 – 1 = 13
Point Estimate = s2 = 26,080,949.45
2.975,13 = 5.00874
2.025,13 = 24.7356
(14  1)( 26,080,949.45)
(14  1)( 26,080,949.45)
≤ 2 ≤
24.7356
5.00874
3,163,167.59 ≤ 2 ≤ 67,692,142.70
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
z 2 2 (2.575) 2 (4.13) 2

E2
12
99% Confidence
= 113.1
Sample 114
c)
E = 10
Range = 500 - 80 = 420
1/4 Range = (.25)(420) = 105
90% Confidence
z.05 = 1.645
z.005 = 2.575
Chapter 8: Statistical Inference: Estimation for Single Populations 16
n =
z 2 2 (1.645) 2 (105) 2

= 298.3
E2
10 2
Sample 299
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
8.41
a)
E = .01
p = .42
96% Confidence
z.02 = 2.05
z 2 p  q ( 2.05) 2 (. 42)(. 58)

= 10237.29
E2
(. 01) 2
Sample size = 10238
n =
b)
E = .03
p = .50
95% Confidence
z.025 = 1.96
z 2 p  q (1.96) 2 (.50)(. 50)
= 1067.11

E2
(.03) 2
Sample size = 1068
n =
c)
E = .04
p = .44 90% Confidence
z.05 = 1.645
z 2 p  q (1.645) 2 (. 44)(. 56)

n =
= 416.73
E2
(. 04) 2
Sample size = 417
d)
E =.02
p = .50
99% Confidence
z 2 p  q (2.575) 2 (. 50)(. 50)

= 4144.14
E2
(. 02) 2
Sample size = 4145
n =
z.005 = 2.575
Chapter 8: Statistical Inference: Estimation for Single Populations 17
8.42
 = $1,000
E = $200
n =
99% Confidence
z.005 = 2.575
z 2 2 (2.575) 2 (1000) 2
= 165.77

E2
200 2
Sample 166
8.43
 = $10.50
E = $3
90% Confidence
z 2 2 (1.645) 2 (10.50) 2
= 33.15

E2
32
Sample size = 34
n =
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,
n =
E = .02
z.05 = 1.645
z 2 p  q (1.645) 2 (. 20)(. 80)

= 1082.41
E2
(. 02) 2
Sample 1083
8.46
p = .50
q = .50
E = .05
z.05 = 1.645
Chapter 8: Statistical Inference: Estimation for Single Populations 18
95% Confidence,
z.025 = 1.96
z 2 p  q (1.96) 2 (.50)(. 50)
n =
= 384.16

E2
(.05) 2
Sample 385
8.47 E = .10
p = .50
q = .50
95% Confidence,
n =
z.025 = 1.96
z 2 p  q (1.96) 2 (.50)(. 50)
= 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
42.55 <  < 48.65
7.75
35
= 45.6 + 3.05
Chapter 8: Statistical Inference: Estimation for Single Populations 19
Chapter 8: Statistical Inference: Estimation for Single Populations 20
8.49
x = 12.03 (point estimate)
/2 = .05
For 90% confidence:
xt
s
n
s = .4373
 12.03  1.833
(.4373)
n = 10
df = 9
t.05,9= 1.833
= 12.03 + .25
10
11.78 <  < 12.28
/2 = .025
For 95% confidence:
xt
s
n
 12.03  2.262
(.4373)
t.025,9 = 2.262
= 12.03 + .31
10
11.72 <  < 12.34
/2 = .005
For 99% confidence:
xt
s
n
 12.03  3.25
(.4373)
t.005,9 = 3.25
= 12.03 + .45
10
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  1.96
= .46 + .0365
n
715
.4235 < p < .4965
b)
n = 284
pˆ  z
p̂ = .71
90% confidence
z.05 = 1.645
pˆ  qˆ
(.71)(.29)
= .71 + .0443
 .71  1.645
n
284
.6657 < p < .7543
Chapter 8: Statistical Inference: Estimation for Single Populations 21
c)
n = 1250
pˆ  z
p̂ = .48
95% confidence
z.025 = 1.96
pˆ  qˆ
(.48)(.52)
 .48  1.96
= .48 + .0277
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
Chapter 8: Statistical Inference: Estimation for Single Populations 22
8.52
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 (1.96) 2 (.70)(. 30)
= 896.4

E2
(.03) 2
Sample 897
z.025 = 1.96
Chapter 8: Statistical Inference: Estimation for Single Populations 23
8.53
n = 17
x = 10.765
/2 = .005
99% confidence
s
xt
n
s = 2.223
 10.765  2.921
2.223
df = 17 - 1 = 16
t.005,16 = 2.921
= 10.765 + 1.575
17
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
196.33 < µ < 229.67
z.01 = 2.33
48
45
= 213 ± 16.67
Chapter 8: Statistical Inference: Estimation for Single Populations 24
8.57
n = 39
90% confidence
xz
 = 3.891
x = 37.256

n
z.05 = 1.645
 37.256  1.645
3.891
= 37.256 ± 1.025
39
36.231 < µ < 38.281
8.58
 = 6
E=1
98% Confidence
z.98 = 2.33
z 2 2 (2.33) 2 (6) 2

= 195.44
E2
12
n =
Sample 196
8.59
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
Chapter 8: Statistical Inference: Estimation for Single Populations 25
8.60
n = 41
x  128
s = 21
98% C.I.
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
s
n
 = 3.06
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
n =
N = 300
z.025 = 1.96
z 2 2 (1.96) 2 (142.50) 2
= 195.02

E2
20 2
Sample 196
df = 41 – 1 = 40
Chapter 8: Statistical Inference: Estimation for Single Populations 26
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
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
95% C.I.
Chapter 8: Statistical Inference: Estimation for Single Populations 27
8.66
n = 27
x = 4.82
95% CI:
s
xt
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
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
n = 560
p̂ =.33
99% Confidence
pˆ  z
z.005= 2.575
pˆ  qˆ
(.33)(.67)
= .33 ± .05
 .33  2.575
n
560
.28 < p < .38
Chapter 8: Statistical Inference: Estimation for Single Populations 28
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
98% confidence
s
xt
n
s = 0.86
/2 = .01
 2.10  2.479
0.86
27
df = 27 - 1 = 26
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 = .0631455
90% C.I.
2.05,22 = 33.9245
(23  1)(.0631455) 2
(23  1)(.0631455) 2
< 2 <
33.9245
12.33801
.0026 < 2 < .0071
8.72
n = 39
xz
x = 1.294

n
 1.294  2.575
1.209 < µ < 1.379
 = 0.205
0.205
39
99% Confidence
= 1.294 ± .085
z.005 = 2.575
Chapter 8: Statistical Inference: Estimation for Single Populations 29
8.73
The sample mean fill for the 58 cans is 11.9788 oz. with a standard deviation of
.0536 oz. The 99% confidence interval for the population fill is 11.9607 oz. to
11.9969 oz. which does not include 12 oz. We are 99% confident that the
population mean is not 12 oz., indicating that the machine may be under filling
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 98%
confident that the actual population mean age of a first-time home buyer is
between 24.0222 years and 31.2378 years.
8.76
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 =
.647887. A 95% confidence interval to estimate the population proportion shows
that we are 95% confident that the actual value lies between .61324 and .681413.
The error of this interval is + .0340865.