08 ch ken black solution

1. 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
s is known.

3. Estimate a population mean from a sample mean when
s 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.

2. 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 when
s 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
s is
known, the z formula can be used even for small samples.

3. 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 (
s 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 (
s 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

4. Chapter 8: Statistical Inference: Estimation for Single Populations 4


Using the Computer to Construct t Confidence Intervals for the
Mean


8.3 Estimating the Population Proportion

Using the Computer to Construct Confidence Inter vals for 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 Point Estimate
Chi-square Distribution Robust
Degrees of Freedom(df) Sample-Size Estimation
Error of Estimation t Distribution
Interval Estimate t Value

5. Chapter 8: Statistical Inference: Estimation for Single Populations 5



SOLUTIONS TO PROBLEMS IN CHAPTER 8


8.1 a) x = 25 s = 3.5 n = 60

95% Confidence z
.025 = 1.96


x + z
n
s
= 25 + 1.96
60
5.3
= 25 +
0.89 = 24.11 < µ < 25.89


b) x = 119.6 s = 23.89 n = 75
98% Confidence z
.01 = 2.33


x + z
n
s
= 119.6 + 2.33
75
89.2
= 119.6 ± 6.43 = 113.17 <
µ < 126.03


c) x = 3.419 s = 0.974 n = 32
90% C.I. z
.05 = 1.645


x + z
n
s
= 3.419 + 1.645
32
974.0
= 3.419 ± .283 =
3.136 <
µ < 3.702


d) x = 56.7 s = 12.1 N = 500 n = 47
80% C.I. z
.10 = 1.28


x ± z
1-
-
N
nN
n
s
= 56.7 + 1.28
1500
47500
47
1.12
-
-
=

56.7 ± 2.15 =
54.55 <
µ < 58.85

6. Chapter 8: Statistical Inference: Estimation for Single Populations 6




8.2 n = 36 x= 211 s = 23
95% C.I. z
.025 = 1.96


x ± z
n
s
= 211 ± 1.96
36
2
= 211 ± 7.51 =
203.49 <
µ < 218.51



8.3 n = 81 x= 47 s = 5.89
90% C.I. z
.05=1.645


x ± z
n
s
= 47 ± 1.645
81
89.5
= 47 ± 1.08 = 45.92 <
µ < 48.08




8.4 n = 70
s
2
= 49
x= 90.4

x = 90.4 Point Estimate

94% C.I. z
.03 = 1.88


x + z
n
s
= 90.4 ± 1.88
70
49
= 90.4 ± 1.57 =
88.83 <
µ < 91.97




8.5
n = 39 N = 200
x = 66 s = 11
96% C.I.
z.02 = 2.05


x ± z
1-
-
N
nN
n
s
= 66 ± 2.05
1200
39200
9
11-
-
=

66 ± 3.25 =
62.75 <
µ < 69.25

x = 66 Point Estimate

7. Chapter 8: Statistical Inference: Estimation for Single Populations 7



8.6 n = 120 x= 18.72 s = 0.8735
99% C.I. z
.005 = 2.575


x = 18.72 Point Estimate

x + z
n
s
= 18.72 ± 2.575
120
8735.0
= 8.72 ± .21 =
18.51 <
µ < 18.93



8.7 N = 1500 n = 187 x= 5.3 years s = 1.28 years
95% C.I. z
.025 = 1.96


x = 5.3 years Point Estimate

x ± z
1-
-N
nN
n
s
= 5.3 ± 1.96
11500
1871500
187
28.1-
-
=

5.3 ± .17 =
5.13 <
µ < 5.47


8.8 n = 24 x= 5.656 s = 3.229
90% C.I. z
.05 = 1.645


x ± z
n
s
= 5.625 ± 1.645
24
23.3
= 5.625 ± 1.085 =
4.540 <
µ < 6.710



8.9 n = 36 x= 3.306 s = 1.17
98% C.I. z
.01 = 2.33


x ± z
n
s
= 3.306 ± 2.33
36
17.1
= 3.306 ± .454 = 2.852 <
µ < 3.760

8. 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. z
.05 = 1.645


x ± z
n
s
= 2.139 ± 1.645
36
)113(.
= 2.139 ± .03 =
2.109 <
µ < 2.169


8.11
m = 27.4 95% confidence interval n = 45


x = 24.533 s = 5.124

z = + 1.96

Confidence interval: x + z
n
s
= 24.533 + 1.96
45
124.5
=

24.533 +
1.497 = 23.036 < mmmm < 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

9. 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
a/2=.025

t
.025,12 = 2.179


n
s
tx±
= 45.62 ± 2.179
13
694.5
= 45.62 ± 3.44 = 42.18 <
µ < 49.06




8.14
n = 12
x= 319.17 s = 9.104 df = 12 - 1 = 11

90% confidence interval


a/2 = .05 t.05,11 = 1.796


n
s
tx±
= 319.17 ± (1.796)

12
104.9
= 319.17 ± 4.72 = 314.45 <
µ < 323.89



8.15
n = 41
x= 128.4 s = 20.64 df = 41 – 1 = 40

98% Confidence Interval

a/2=.01


t.01,40 = 2.423


n
s
tx±
= 128.4 ± 2.423
27
6.20
= 128.4 ± 9.61 = 118.79 <
µ < 138.01

x = 128.4 Point Estimate

10. Chapter 8: Statistical Inference: Estimation for Single Populations 10




8.16 n = 15 x= 2.364 s
2
= 0.81 df = 15 – 1 = 14

90% Confidence interval
a/2=.05

t
.05,14 = 1.761


n
s
tx±
= 2.364 ± 1.761
15
81.0
= 2.364 ± .409 =
1.955 <
µ < 2.773



8.17
n = 25
x = 16.088 s = .817 df = 25 – 1 = 24

99% Confidence Interval


a/2=.005


t.005,24 = 2.797


n
s
tx±
= 16.088 ± 2.797
25
817.
= 16.088 ± .457 =
15.631 <
µ < 16.545

x = 16.088 Point Estimate




8.18
n = 22
x = 1,192 s = 279 df = n - 1 = 21

98% CI and
a/2 = .01 t.01,21 = 2.518


n
s
tx±
= 1,192 +
(2.518)
22
279
= 1,192 + 149.78 = 1,042.22 < mmmm < 1,341.78

11. Chapter 8: Statistical Inference: Estimation for Single Populations 11



8.19 n = 20 df = 19 95% CI t
.025,19 = 2.093


x = 2.36116 s = 0.19721

2.36116 + 2.093
20
1972.0
= 2.36116 + 0.0923 = 2.26886 < mmmm < 2.45346

Point Estimate =
2.36116

Error =
0.0923




8.20
n = 28
x= 5.335 s = 2.016 df = 28 – 1 = 27

90% Confidence Interval
a/2=.05


t.05,27 = 1.703


n
s
tx±
= 5.335 ± 1.703
28
016.2
= 5.335 + .649 = 4.686 < µ < 5.984



8.21
n = 10
x= 49.8 s = 18.22 df = 10 – 1 = 9

95% Confidence
a/2=.025 t.025,9 = 2.262


n
s
tx±
= 49.8 ± 2.262
10
22.18
= 49.8 +
13.03 = 36.77 < µ < 62.83

12. Chapter 8: Statistical Inference: Estimation for Single Populations 12




8.22 n = 14, 98% confidence, a/2 = .01, df = 13

t
.01,13 = 2.650

from data:
x = 152.16 s = 14.42

confidence interval:
n
s
tx±
= 152.16 +
2.65
14
42.14
=

152.16 +
10.21 = 141.95 < mmmm < 162.37


The point estimate is 152.16



8.23 a)
n = 44 p
ˆ=.51 99% C.I. z .005 = 2.575


n
qp
zp
ˆˆ
ˆ
×
± = .51 ± 2.575
44
)49)(.51(.
= .51 ± .194 =
.316 <
p< .704


b) n = 300
p
ˆ= .82 95% C.I. z .025 = 1.96


n
qp
zp
ˆˆ
ˆ
×
±
= .82 ± 1.96
300
)18)(.82(.
= .82 ± .043 = .777 <
p < .863


c) n = 1150
p
ˆ= .48 90% C.I. z .05 = 1.645


n
qp
zp
ˆˆ
ˆ
×
±
= .48 ± 1.645
1150
)52)(.48(.
= .48 ± .024 = .456 <
p < .504


d) n = 95
p
ˆ= .32 88% C.I. z .06 = 1.555


n
qp
zp
ˆˆ
ˆ
×
±
= .32 ± 1.555
95
)68)(.32(.
= .32 ± .074 = .246 <
p < .394

13. Chapter 8: Statistical Inference: Estimation for Single Populations 13




8.24 a) n = 116 x = 57 99% C.I. z
.005 = 2.575


p
ˆ =

116
57
=
n
x

= .49


n
qp
zp
ˆˆ
ˆ
×
±
= .49 ± 2.575
116
)51)(.49(.
= .49 ± .12 = .37 <
p < .61


b) n = 800 x = 479 97% C.I. z
.015 = 2.17


p
ˆ =
800
479
=
n
x
= .60


n
qp
zp
ˆˆ
ˆ
×
±
= .60 ± 2.17
800
)40)(.60(.
= .60 ± .038 = .562 <
p < .638


c) n = 240 x = 106 85% C.I. z
.075 = 1.44


p
ˆ =
240
106
=
n
x
= .44


n
qp
zp
ˆˆ
ˆ
×
±
= .44 ± 1.44
240
)56)(.44(.
= .44 ± .046 = .394 <
p < .486


d) n = 60 x = 21 90% C.I. z
.05 = 1.645


p
ˆ =
60
21
=
n
x
= .35


n
qp
zp
ˆˆ
ˆ
×
±
= .35 ± 1.645
60
)65)(.35(.
= .35 ± .10 = .25 <
p < .45

14. Chapter 8: Statistical Inference: Estimation for Single Populations 14




8.25 n = 85 x = 40 90% C.I. z
.05 = 1.645


p
ˆ

=
85
40
=
n
x
= .47


n
qp
zp
ˆˆ
ˆ
×
±
= .47 ± 1.645
85
)53)(.47(.
= .47 ± .09 = .38 <
p < .56



95% C.I. z
.025 = 1.96


n
qp
zp
ˆˆ
ˆ
×
±
= .47 ± 1.96
85
)53)(.47(.
= .47 ± .106 = .364 <
p < .576



99% C.I. z
.005 = 2.575


n
qp
zp
ˆˆ
ˆ
×
±
= .47 ± 2.575
85
)53)(.47(.
= .47 ± .14 = .33 <
p < .61


All things being constant, as the confidence increased, the width of the interval
increased.




8.26 n = 1003
p
ˆ = .245 99% CI z .005 = 2.575


n
qp
zp
ˆˆ
ˆ
×
±
= .245 +
2.575
1003
)755)(.245(.
= .245 +
.035 = .21 < p < .28

15. Chapter 8: Statistical Inference: Estimation for Single Populations 15


8.27 n = 560
p
ˆ = .47 95% CI z.025 = 1.96


n
qp
zp
ˆˆ
ˆ
×
±
= .47 +
1.96
560
)53)(.47(.
= .47 +
.0413 = .4287 < p < .5113


n = 560
p
ˆ = .28 90% CI z .05 = 1.645


n
qp
zp
ˆˆ
ˆ
×
±
= .28 +
1.645
560
)72)(.28(.
= .28 +
.0312 = .2488 < p < .3112




8.28 n = 1250 x = 997 98% C.I. z
.01 = 2.33


p
ˆ=
1250
997
=
n
x
= .80


n
qp
zp
ˆˆ
ˆ
×
±
= .80 ± 2.33
1250
)20)(.80(.
= .80 ± .026 = .774 <
p < .826




8.29 n = 3481 x = 927


p
ˆ=
3481
927
=
n
x
= .266


a)
p
ˆ= .266 Point Estimate


b) 99% C.I. z
.005 = 2.575


n
qp
zp
ˆˆ
ˆ
×
±
= .266 +
2.575
3481
)734)(.266(.
= .266 ± .02 =

.246 <
p < .286

16. Chapter 8: Statistical Inference: Estimation for Single Populations 16



8.30 n = 89 x = 48 85% C.I. z
.075 = 1.44


p
ˆ =
89
48
=
n
x
= .54


n
qp
zp
ˆˆ
ˆ
×
±
= .54 ± 1.44
89
)46)(.54(.
= .54 ± .076 = .464 <
p < .616



8.31
p
ˆ = .63 n = 672 95% Confidence z = + 1.96


n
qp
zp
ˆˆ
ˆ
×
±
= .63 +
1.96
672
)37)(.63(.
= .63 +
.0365 = .5935 < p < .6665




8.32 a) n = 12 x = 28.4 s
2
= 44.9 99% C.I. df = 12 – 1 = 11

c
2
.995,11
= 2.60321 c
2
.005,11
= 26.7569


7569.26
)9.44)(112(-
< s
2
<
60321.2
)9.44)(112(-



18.46 <
s
2
< 189.73


b)
n = 7
x = 4.37 s = 1.24 s
2
= 1.5376
95% C.I. df = 12 – 1 = 11

c
2
.975,6
= 1.237347 c
2
.025,6
= 14.4494


4494.14
)5376.1)(17(-
< s
2
<
237347.1
)5376.1)(17(-



0.64 <
s
2
< 7.46


c)
n = 20
x = 105 s = 32 s
2
= 1024
90% C.I. df = 20 – 1 = 19

17. Chapter 8: Statistical Inference: Estimation for Single Populations 17


c
2
.95,19
= 10.117 c
2
.05,19
= 30.1435


1435.30
)1024)(120(-
< s
2
<
117.10
)1024)(120(-



645.45 <
s
2
< 1923.10


d)
n = 17 s
2
= 18.56 80% C.I. df = 17 – 1 = 16

c
2
.90,16
= 9.31223 c
2
.10,16
= 23.5418


5418.23
)56.18)(117(-
< s
2
<
31223.9
)56.18)(117(-



12.61 <
ssss
2
< 31.89

8.33
n = 16 s
2
= 37.1833 98% C.I. df = 16-1 = 15

c
2
.99,15
= 5.22935 c
2
.01,15
= 30.5779


5779.30
)1833.37)(116(-
< s
2
<
22935.5
)1833.37)(116(-



18.24 <
ssss
2
< 106.66


8.34
n = 12 s = 4.3 s
2
= 18.49
98% C.I. df = 20 – 1 = 19

c
2
.99,19
= 7.63273 c
2
.01,19
= 36.1980


1980.36
)49.18)(120(-
< s
2
<
63273.7
)49.18)(120(-



9.71 <
s
2
< 46.03


Point Estimate = s
2
= 18.49


8.35
n = 152 s
2
= 3.067 99% C.I. df = 15 – 1 = 14

c
2
.995,14
= 4.07468 c
2
.005,14
= 31.3193

18. Chapter 8: Statistical Inference: Estimation for Single Populations 18



3193.31
)067.3)(115(-
< s
2
<
07468.24
)067.3)(115(-



1.37 <
s
2
< 10.54


8.36
n = 14 s
2
= 26,798,241.76 95% C.I. df = 14 – 1 = 13

Point Estimate =
s
2
= 26,798,241.76


c
2
.975,13
= 5.00874 c
2
.025,13
= 24.7356


7356.24
)76.241,798,26)(114(-
< s
2
<
00874.5
)76.241,798,26)(114(-



14,084,038.51 <
s
2
< 69,553,848.45


8.37 a)
s = 36 E = 5 95% Confidence z.025 = 1.96


n =
2
22
2
22
5
)36()96.1(
=
E
zs
= 199.15

Sample 200


b)
s = 4.13 E = 1 99% Confidence z .005 = 2.575

n =
2
22
2
22
1
)13.4()575.2(
=
E
zs
= 113.1


Sample 114


c) E = 10 Range = 500 - 80 = 420

1/4 Range = (.25)(420) = 105

90% Confidence z
.05 = 1.645

n =
2
22
2
22
10
)105()645.1(
=
E
zs
= 298.3


Sample 299

19. 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 z
.06 = 1.555

n =
2
22
2
22
3
)5.14()555.1(
=
E
zs
= 56.5


Sample 57



8.38 a) E = .02 p=.40 96% Confidence z
.02 = 2.05

n =
2
2
2
2
)02(.
)60)(.40(.)05.2(
=
×
E
qpz
= 2521.5


Sample 2522


b) E = .04 p=.50 95% Confidence z
.025 = 1.96

n =
2
2
2
2
)04(.
)50)(.50(.)96.1(
=
×
E
qpz
= 600.25


Sample 601



c) E = .05 p = .55 90% Confidence z
.05 = 1.645

n =
2
2
2
2
)05(.
)45)(.55(.)645.1(
=
×
E
qpz
= 267.9


Sample 268

20. Chapter 8: Statistical Inference: Estimation for Single Populations 20



d) E =.01 p = .50 99% Confidence z
.005 = 2.575

n =
2
2
2
2
)01(.
)50)(.50(.)575.2(
=
×
E
qpz
= 16,576.6

Sample 16,577



8.39 E = $200
s = $1,000 99% Confidence z .005 = 2.575

n =
2
22
2
22
200
)1000()575.2(
=
E
zs
= 165.77


Sample 166



8.40 E = $2
s = $12.50 90% Confidence z .05 = 1.645

n =
2
22
2
22
2
)50.12()645.1(
=
E
zs
= 105.7


Sample 106



8.41 E = $100 Range = $2,500 - $600 = $1,900


s » 1/4 Range = (.25)($1,900) = $475

90% Confidence z
.05 = 1.645

n =
2
22
2
22
100
)475()645.1(
=
E
zs
= 61.05


Sample 62

21. Chapter 8: Statistical Inference: Estimation for Single Populations 21




8.42 p = .20 q = .80 E = .02

90% Confidence, z
.05 = 1.645

n =
2
2
2
2
)02(.
)80)(.20(.)645.1(
=
×
E
qpz
= 1082.41

Sample 1083



8.43 p = .50 q = .50 E = .05

95% Confidence, z
.025 = 1.96

n =
2
2
2
2
)05(.
)50)(.50(.)96.1(
=
×
E
qpz
= 384.16

Sample 385



8.44 E = .10 p = .50 q = .50

95% Confidence, z
.025 = 1.96

n =
2
2
2
2
)10(.
)50)(.50(.)96.1(
=
×
E
qpz
= 96.04

Sample 97

22. Chapter 8: Statistical Inference: Estimation for Single Populations 22




8.45 x = 45.6 s = 7.75 n = 35

80% confidence z
.10 = 1.28


35
75.7
28.16.45±=±
n
zx
s
= 45.6 + 1.677


43.923 <
mmmm < 47.277


94% confidence z
.03 = 1.88


35
75.7
88.16.45±=±
n
zx
s
= 45.6 + 2.463


43.137 <
mmmm < 48.063


98% confidence z
.01 = 2.33


35
75.7
33.26.45±=±
n
zx
s
= 45.6 + 3.052


42.548 <
mmmm < 48.652



8.46 x = 12.03 (point estimate) s = .4373 n = 10 df = 9

For 90% confidence:
a/2 = .05 t .05,9= 1.833


10
4373.
833.103.12
±=±
n
s
tx = 12.03 +
.25


11.78 <
mmmm < 12.28

23. Chapter 8: Statistical Inference: Estimation for Single Populations 23



For 95% confidence: a/2 = .025 t
.025,9 = 2.262


10
4373.
262.203.12±=±
n
s
tx = 12.03 + .31


11.72 <
mmmm < 12.34


For 99% confidence: a/2 = .005 t
.005,9 = 3.25


10
)4373(.
25.303.12±=±
n
s
tx = 12.03 + .45


11.58 <
mmmm < 12.48




8.47 a) n = 715 x = 329


715
329
ˆ=p = .46

95% confidence z
.025 = 1.96


715
)54)(.46(.
96.146.
ˆˆ
ˆ ±=
×
±
n
qp
zp = .46 + .0365


.4235 <
p < .4965


b) n = 284
p
ˆ = .71 90% confidence z .05 = 1.645


284
)29)(.71(.
645.171.
ˆˆ
ˆ ±=
×
±
n
qp
zp
= .71 +
.0443


.6657 <
p < .7543

24. Chapter 8: Statistical Inference: Estimation for Single Populations 24




c) n = 1250
p
ˆ = .48 95% confidence z .025 = 1.96


1250
)52)(.48(.
96.148.
ˆˆ
ˆ ±=
×
±
n
qp
zp
= .48 +
.0277

.4523 < p < .5077


d) n = 457 x = 270 98% confidence z
.01 = 2.33


457
270
ˆ=p
= .591

457
)409)(.591(.
33.2591.
ˆˆ
ˆ ±=
×
±
n
qp
zp
= .591 +
.0536

.5374 < p < .6446



8.48 n = 10 s = 7.40045 s
2
= 54.7667 df = 10 – 1 = 9

90% confidence, a/2 = .05 1 - a/2 = .95

c
2
.95,9
= 3.32511 c
2
.05,9
= 16.919


919.16
)7667.54)(110(-
< s
2
<
32511.3
)7667.54)(110(-



29.133 <
s
2
< 148.236


95% confidence, a/2 = .025 1 - a/2 = .975

c
2
.975,9
= 2.70039 c
2
.025,9
= 19.0228


0228.19
)7667.54)(110(-
< s
2
<
70039.2
)7667.54)(110(-



4.258 <
s
2
< 182.529

25. Chapter 8: Statistical Inference: Estimation for Single Populations 25




8.49 a)
s = 44 E = 3 95% confidence z
.025 = 1.96

n =
2
22
2
22
3
)44()96.1(
=
E
zs
= 826.4

Sample 827


b) E = 2 Range = 88 - 20 = 68

use = 1/4(range) = (.25)(68) = 17

90% confidence z
.05 = 1.645


2
22
2
22
2
)17()645.1(
=
E
zs
= 195.5


Sample 196


c) E = .04 p = .50 q = .50

98% confidence z
.01 = 2.33


2
2
2
2
)04(.
)50)(.50(.)33.2(
=
×
E
qpz
= 848.3


Sample 849


d) E = .03 p = .70 q = .30

95% confidence z
.025 = 1.96



2
2
2
2
)03(.
)30)(.70(.)96.1(4
=
×
E
qpz
= 896.4

Sample 897

26. Chapter 8: Statistical Inference: Estimation for Single Populations 26




8.50 n = 17 x= 10.765 S = 2.223 df = 17 - 1 = 16

99% confidence a/2 = .005 t
.005,16 = 2.921


17
223.2
921.2765.10±=±
n
s
tx = 10.765 + 1.575

9.19 <
µ < 12.34



8.51 p=.40 E=.03 90% Confidence z
.05 = 1.645

n =
2
2
2
2
)03(.
)60)(.40(.)645.1(
=
×
E
qpz

= 721.61


Sample 722



8.52 n = 17 s
2
= 4.941 99% C.I. df = 17 – 1 = 16

c
2
.995,16
= 5.14224 c
2
.005,16
= 34.2672


2672.34
)941.4)(117(-
< s
2
<
14224.5
)941.4)(117(-



2.307 <
s
2
< 15.374



8.53 n = 45 x= 213 s = 48

98% Confidence z
.01 = 2.33


45
48
33.2213±=±
n
zx
s
= 213 ± 16.67


196.33 <
µ < 229.67

27. 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


39
891.3
645.1256.37±=±
n
zx
s
= 37.256 ± 1.025


36.231 <
µ < 38.281



8.55
s = 6 E=1 98% Confidence z .98 = 2.33

n =
2
22
2
22
1
)6()33.2(
=
E
zs
= 195.44


Sample 196



8.56 n = 1,255 x = 714 95% Confidence z
.025 = 1.96


1255
714
ˆ=p
= .569

255,1
)431)(.569(.
96.1569.
ˆˆ
ˆ ±=
×
±
n
qp
zp = .569 ± .027


.542 <
p < .596

28. Chapter 8: Statistical Inference: Estimation for Single Populations 28



8.57 n = 41 s = 21 x = 128 98% C.I. df = 41 – 1 = 40

t
.01,40 = 2.423

Point Estimate = $128


25
21
423.2128±=±
n
s
tx = 128 + 7.947


120.053 <
mmmm < 135.947


Interval Width = 135.947 – 120.053 = 15.894




8.58 n = 60
x = 6.717 s = 3.06 N =300

98% Confidence z
.01 = 2.33


1300
60300
60
06.3
33.2717.6
1-
-
±=
-
-
±N
nN
n
zx
s
=


6.717 ± 0.825 =


5.892 <
µ < 7.542



8.59 E = $20 Range = $600 - $30 = $570

1/4 Range = (.25)($570) = $142.50

95% Confidence z
.025 = 1.96

n =
2
22
2
22
20
)50.142()96.1(
=
E
zs
= 195.02

Sample 196

29. Chapter 8: Statistical Inference: Estimation for Single Populations 29




8.60 n = 245 x = 189 90% Confidence z
.05= 1.645


245
189
ˆ ==
n
x
p = .77


245
)23)(.77(.
645.177.
ˆˆ
ˆ ±=
×
±
n
qp
zp
= .77 ± .044


.726 <
p < .814




8.61 n = 90 x = 30 95% Confidence z
.025 = 1.96


90
30
ˆ==
n
x
p = .33

90
)67)(.33(.
96.133.
ˆˆ
ˆ ±=
×
±
n
qp
zp
= .33 ± .097


.233 <
p < .427



8.62 n = 12 x = 43.7 s
2
= 228 df = 12 – 1 = 11 95% C.I.

t
.025,11 = 2.201


12
228
201.27.43
±=±
n
s
tx = 43.7 +
9.59


34.11 <
mmmm < 53.29



c
2
.975,11
= 3.81575 c
2
.025,11
= 21.92


92.21
)228)(112(-
< s
2
<
81575.3
)228)(112(-



114.42 <
s
2
< 657.28

30. Chapter 8: Statistical Inference: Estimation for Single Populations 30




8.63 n = 27 x = 4.82 s = 0.37 df = 26

95% CI: t
.025,26 = 2.056


27
37.0
056.282.4±=±
n
s
tx = 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 z
.025 = 1.96


77
12
96.148.2±=±
n
zx
s
= 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 n = 560
p
ˆ=.33

99% Confidence z
.005= 2.575


560
)67)(.33(.
575.233.
ˆˆ
ˆ ±=
×
±
n
qp
zp
= .33 ± (2.575) = .33 ± .05


.28 <
p < .38

31. Chapter 8: Statistical Inference: Estimation for Single Populations 31



8.66 p = .50 E = .05 98% Confidence z
.01 = 2.33


2
2
2
2
)05(.
)50)(.50(.)33.2(
=
×
E
qpz
= 542.89

Sample 543




8.67 n = 27
x= 2.10 s = 0.86 df = 27 - 1 = 26

98% confidence a/2 = .01 t
.01,26 = 2.479


27
86.0
479.210.2±=±
n
s
tx = 2.10 ± (2.479) = 2.10 ± 0.41


1.69 <
µ < 2.51




8.68 n = 23 df = 23 – 1 = 22 s = .0631455 90% C.I.

c
2
.95,22
= 12.338 c
2
.05,22
= 33.9244


9244.33
)0631455)(.123(
2
-
< s
2
<
338.12
)0631455)(.123(
2
-



.0026 <
s
2
< .0071




8.69 n = 39 x= 1.294 s = 0.205 99% Confidence z .005 = 2.575


39
205.0
575.2294.1±=±
n
zx
s
= 1.294 ± (2.575) = 1.294 ± .085


1.209 <
µ < 1.379

32. 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.