CHAPTER
Confidence Intervals
223
6
6.1 CONFIDENCE INTERVALS FOR THE MEAN (LARGE SAMPLES)
6.1 Try It Yourself Solutions
2a. 1.96, 30, s 51.0
c
zn==≈
b. 51.0
1.96 18.3
30
c
s
Ez n
=≈ ≈
c. You are 95% confident that the maximum error of the estimate is about 18.3 friends.
4b. 75% CI: (121.2, 140.4)
85% CI: (118.7, 142.9)
99% CI: (109.2, 152.4)
c. As the confidence level increases, so does the width of the interval.
6a. 1.96, 10, 53.0
c
zE
σ
==≈
b.
22
1.96 53.0 107.91 108
10
c
z
nE
σ
⋅
⎛⎞
⎛⎞
=≈ ≈→
⎜⎟
⎜⎟
⎝⎠
⎝⎠
c. You should have at least 108 users in you sample. Because of the larger margin of error, the
sample size needed is much smaller.
224 CHAPTER 6 │ CONFIDENCE INTERVALS
6.1 EXERCISE SOLUTIONS
1. You are more likely to be correct using an interval estimate because it is unlikely that a point
estimate will equal the population mean exactly.
3. d; As the level of confidence increases, c
zincreases therefore creating wider intervals.
4. No, the 95% confidence interval means that with 95% confidence you can say that the population
5. 1.28 6. 1.44 7. 1.15 8. 2.17
9. 3.8 4.27 0.47x
µ
−= − =− 10. 9.5 8.76 0.74x
µ
−= − =
µ
µ
17. 0.88 1.555
c
cz=⇒=
57.2, 7.1, 50xsn===
7.1
57.2 1.555 57.2 1.561 (55.6, 58.8)
50
c
s
xz n
±=± ≈±≈
Answer: (c)
19. 0.95 1.96
c
cz=⇒=
57.2, 7.1, 50xsn===
CHAPTER 6 │ CONFIDENCE INTERVALS 225
21. 1.5
12.3 1.645 12.3 0.349 (12.0, 12.6)
50
c
s
xz n
±=± ≈±≈
22. 0.8
31.39 1.96 31.3 0.173 (31.22, 31.56)
82
c
s
xz n
±=± ≈±≈
25. (12.0, 14.8) 13.4 1.4 13.4,x⇒±⇒= E = 1.4
26. (21.61, 30.15) 25.88 4.27 25.88,x⇒±⇒= E = 4.27
30. 0.95 1.96
c
cz=⇒=
22
(1.96)(2.5) 24.01 25
1
c
z
nE
σ
⎛⎞
⎛⎞
== ≈⇒
⎜⎟
⎜⎟
⎝⎠
⎝⎠
31. 0.80 1.28
c
cz=⇒=
22
(1.28)(4.1) 6.89 7
2
c
z
nE
σ
⎛⎞
⎛⎞
== ≈⇒
⎜⎟
⎜⎟
⎝⎠
⎝⎠
226 CHAPTER 6 │ CONFIDENCE INTERVALS
33. (26.2, 30.1) 2 30.1 26.2 3.9 1.95EE⇒= − =⇒= and x = 26.2 + E
= 26.2 + 1.95 = 28.15
34. (44.07, 80.97) 2 80.97 44.07 36.9 18.45EE⇒= − = ⇒= and x = 44.07 + E
= 44.07 + 18.45 = 62.52
36. 90% CI: 0.32
2.34 1.645 2.34 0.076 (2.26, 2.42)
48
c
s
xz n
±=± ≈±≈
95% CI: 0.32
2.34 1.96 2.34 0.091 (2.25, 2.43)
48
c
s
xz n
±=± ≈±≈
With 90% confidence, you can say that the population mean price is between $2.26 and $2.42
with 95% confidence, you can say that the population mean price is between $2.25 and $2.43.
The 95% CI is wider.
38. 90% CI: 6.7
23 1.645 23 1.837 (21, 25)
36
c
s
xz n
±=± ≈±≈
95% CI: 6.7
23 1.96 23 2.189 (21, 25)
36
c
s
xz n
±=± ≈±≈
With 90% confidence and with 95% confidence, you can say that the population mean
concentration is between 21 and 25 cubic centimeters per cubic meter, when rounded to the
nearest whole number, both confidence intervals have the same width.
CHAPTER 6 │ CONFIDENCE INTERVALS 227
40. 15.5
150 2.575 150 5.15 (144.85, 155.15)
60
c
s
xz n
±=± ≈±≈
With 99% confidence, you can say that the population mean repair cost is between $144.85 and
$155.15.
42. 15.5
150 2.575 150 6.31 (143.69, 156.31)
40
c
s
xz n
±=± ≈±≈
The n = 40 CI is wider because a smaller sample is taken, giving less information about the
population.
44. 38.60
154.17 2.575 154.17 13.40 (140.77, 167.57)
55
c
s
xz n
±= ± ≈ ±≈
With 99% confidence, you can say that the population mean nightly cost is between $140.77 and
$167.57.
46. 42.50
154.17 2.575 154.17 14.76 (139.41, 168.93)
55
c
s
xz n
±= ± ≈ ±≈
The s = 42.50 CI is wider because of the increased variability within the sample.
228 CHAPTER 6 │ CONFIDENCE INTERVALS
47. (a) An increase in the level of confidence will widen the confidence interval.
48. Answers will vary.
49. 302 15.1
20
x
xn
==≈
∑
50. 90% CI: 4.5
29 1.645 29 1.425 (27.6, 30.4)
27
c
xz n
σ
±=± ≈±≈
99% CI: 4.5
29 2.575 29 2.230 (26.8, 31.2)
27
c
xz n
σ
±=± ≈±≈
With 90% confidence, you can say that the population mean length of time is between 27.6 and
30.4 minutes. With 99% confidence, you can say that the population mean length of time is
between 26.8 and 31.2 minutes.
The 99% CI is wider.
53. (a)
22
1.96 2.8 120.473 121
0.5
c
z
nE
σ
⋅
⎛⎞
⎛⎞
== ≈→
⎜⎟
⎜⎟
⎝⎠
⎝⎠ servings
CHAPTER 6 │ CONFIDENCE INTERVALS 229
54. (a)
22
1.645 1.2 3.897 4
1
c
z
nE
σ
⋅
⎛⎞
⎛⎞
== ≈→
⎜⎟
⎜⎟
⎝⎠
⎝⎠ students
(b)
22
2.575 1.2 9.548 10
1
c
z
nE
σ
⋅
⎛⎞
⎛⎞
== ≈→
⎜⎟
⎜⎟
⎝⎠
⎝⎠ students
The 99% CI requires a larger sample because more information is needed from the population
to be 99% confident.
56. (a)
22
1.96 3 34.574 35
1
c
z
nE
σ
⋅
⎛⎞
⎛⎞
== ≈→
⎜⎟
⎜⎟
⎝⎠
⎝⎠ bottles
(b)
22
1.96 3 8.644 9
2
c
z
nE
σ
⋅
⎛⎞
⎛⎞
== ≈→
⎜⎟
⎜⎟
⎝⎠
⎝⎠ bottles
E = 1 requires a larger sample size. As the error size decreases, a larger sample must be taken
to obtain enough information from the population to ensure the desired accuracy.
58. (a)
22
1.645 0.15 33.708 34
0.0425
c
z
nE
σ
⋅
⎛⎞
⎛⎞
== ≈→
⎜⎟
⎜⎟
⎝⎠
⎝⎠ units
(b)
22
1.645 0.15 134.833 135
0.02125
c
z
nE
σ
⋅
⎛⎞
⎛⎞
== ≈→
⎜⎟
⎜⎟
⎝⎠
⎝⎠ units
230 CHAPTER 6 │ CONFIDENCE INTERVALS
60. (a)
22
2.575 0.20 11.788 12
0.15
c
z
nE
σ
⋅
⎛⎞
⎛⎞
== ≈→
⎜⎟
⎜⎟
⎝⎠
⎝⎠ soccer balls
(b)
22
2.575 0.10 2.947 3
0.15
c
z
nE
σ
⋅
⎛⎞
⎛⎞
== ≈→
⎜⎟
⎜⎟
⎝⎠
⎝⎠ soccer balls
σ
= 0.2 requires a larger sample size. Due to the increased variability in the population, a
larger sample is needed to ensure the desired accuracy.
62. Sample answer: A 99% CI may not be practical to use in all situations. It may produce a CI so
wide that is has no practical application.
63. (212.74, 221.51)
With 95% confidence, you can say that the population mean airfare price is between $212.74 and
$221.51.
65. 80% confidence interval results:
:
µ
population mean
standard deviation = 344.9
mean n Sample mean Std. err. L. Limit U Limit
µ
30 1042.7 62.969837 962.0009 1123.399
µ
CHAPTER 6 │ CONFIDENCE INTERVALS 231
µ
µ
66. 80% confidence interval results:
:
µ
mean of variable
standard deviation not specified
Variable N Sample mean Std. err. L. Limit U Limit
Carbohydrate (grams) 30 41.966667 2.1493027 39.212223 44.721107
90% confidence interval results:
:
µ
mean of variable
standard deviation not specified
µ
With 80% confidence, you can say that the population mean carbohydrate content is between 39.2
and 44.7 grams. With 90% confidence, you can say it is between 38.4 and 45.5 grams. With 95%
confidence, you can say it is between 37.8 and 46.2 grams.
67. (a) 1000 500 0.707
1 1000 1
Nn
N
−−
=≈
−−
(b) 1000 100 0.949
1 1000 1
Nn
N
−−
=≈
−−
232 CHAPTER 6 │ CONFIDENCE INTERVALS
68. (a) 100 50 0.711
11001
Nn
N
−−
=≈
−− (b) 400 50 0.937
14001
Nn
N
−−
=≈
−−
69. Sample answer:
Write original equation.
c
z
En
σ
=
6.2 CONFIDENCE INTERVALS FOR THE MEAN
(SMALL SAMPLES)
6.2 Try It Yourself Solutions
1a. d.f. = n − 1 = 22 − 1 = 21
b. c = 0.90
c. 1.721
c
t=
2a. 90% CI: 1.753
c
t=
10
1.753 4.4
16
c
s
Et n
== ≈
CHAPTER 6 │ CONFIDENCE INTERVALS 233
3a. 90% CI: 1.729
c
t=
2.39
1.729 0.92
20
c
s
Et n
== ≈
95% CI: 2.093
c
t=
2.39
2.093 1.12
c
s
Et n
=− ≈
6.2 EXERCISE SOLUTIONS
1. 1.833
c
t= 2. 2.201
c
t= 3. 2.947
c
t= 4. 2.539
c
t=
5. 5
2.131 2.7
16
c
s
Et n
== ≈ 6. 3
4.032 4.9
6
c
s
Et n
== ≈
10. (a) 0.85
13.4 2.365 13.4 0.711 (12.7, 14.1)
8
c
s
xt n
±=± ≈±≈
(b) 0.85
13.4 1.96 13.4 0.589 (12.8, 14.0)
8
c
s
xz n
±=± ≈±≈
The t-CI is wider.
234 CHAPTER 6 │ CONFIDENCE INTERVALS
11. (a) 0.34
4.3 2.650 4.3 0.241 (4.1, 4.5)
14
c
s
xt n
±=± ≈±≈
12. (a) 4.6
24.7 3.250 24.7 4.728 (20.0, 29.4)
10
c
s
xt n
±=± ≈±≈
13. (14.7, 22.1) 18.4 22.1 18.4 3.7xE⇒= ⇒= − =
14. (6.17, 8.53) 7.35 8.53 7.35 1.18xE⇒= ⇒= − =
18. 5.8
2.776 7.2
5
c
s
Et n
== ≈
22.2 7.2 (15.0, 29.4)xE±≈ ± =
With 95% confidence, you can say that the population mean driving distance to work is between
15.0 and 29.4 miles.
CHAPTER 6 │ CONFIDENCE INTERVALS 235
22. (a) 0.28
1.50 1.796 1.50 0.145 (1.36, 1.65)
12
c
s
xt n
±=± ≈±≈
(b) 0.28
1.50 1.645 1.50 0.019 (1.48, 1.52)
600
c
s
xz n
±=± ≈±≈
The t-CI is wider.
24. (a) 68,555.6x≈ (b) s ≈ 3243.5
(c) 3243.5
68,555.6 3.012 68,555.6 2611.0 (65,944.6, 71,166.6)
14
c
s
xt n
±= ± ≈ ± ≈
26. (a) 2.35x≈
(b) 1.03s≈
(c) 1.03
2.35 2.977 2.35 0.792 (1.56, 3.14)
15
c
s
xt n
±≈± ≈±≈
28. 4.99x≈, s = 0.36, n < 30,
σ
known, and pop normally distributed → use t–distribution
0.36
4.99 2.145 4.99 0.20 (4.79, 5.19)
15
c
s
xt n
±=± ≈±=
With 95% confidence, you can say that the population mean interest rate is between 4.79% and
5.19%.
236 CHAPTER 6 │ CONFIDENCE INTERVALS
30. 4.3x=, s = 1.14, n < 30,
σ
known, and pop normally distributed → use normal distribution
1.34
4.3 1.96 4.3 0.59 (3.7, 4.9)
20
c
xz n
σ
±≈± ≈±≈
With 95% confidence, you can say that the population mean is between 3.7 and 4.9 yards per
carry.
33. 90% confidence interval results:
:
µ
mean of variable
Variable Sample mean Std. err. DF L. Limit U Limit
Time (in hours) 12.194445 0.4136141 17 11.474918 12.91397
95% confidence interval results:
:
µ
mean of variable
µ
CHAPTER 6 │ CONFIDENCE INTERVALS 237
34. 90% confidence interval results:
:
µ
population mean
Mean Sample mean Std. err. DF L. Limit U Limit
µ
7.2 0.57287157 10 6.1616926 8.238307
µ
µ
µ
µ
With 90% confidence, you can say the population mean weekly time spent weightlifting is
between 6.2 and 8.2 hours. With 95% confidence, you can say it is between 5.9 and 8.5 hours.
With 99% confidence, you can say it is between 5.4 and 9.0 hours. As the level of confidence
increases, the intervals get wider.
35. n = 25, 56.0x=, s = 0.25
0.99 99%t±→ t-CI
0.25
56.0 2.797 56.0 0.140 (55.9, 56.1)
25
c
s
xt n
±=± ≈±≈
6.3 CONFIDENCE INTERVALS FOR POPULATION PROPORTIONS
6.3 Try It Yourself Solutions
1a. x = 181, n = 1006
b. 181
ˆ0.180
1006
p=≈
238 CHAPTER 6 │ CONFIDENCE INTERVALS
3a. n = 498, ˆ0.25p=
ˆˆ
1 1 0.25 0.75qp== −= ≈
b. ˆ498 0.25 124.5 5np =⋅ = >
ˆ498 0.75 373.5 5nq =⋅ = >
Distribution of
ˆ
p is approximately normal.
4a. (1) ˆˆ
0.5, 0.5pq==
1.645, 0.02
c
zE==
(2)
ˆˆ
0.11, 0.89pq==
1.645, 0.02
c
zE==
6.3 EXERCISE SOLUTIONS
1. False. To estimate the value of p, the population proportion of successes, use the point estimate
ˆ.
x
pn
=
CHAPTER 6 │ CONFIDENCE INTERVALS 239
3. 752
ˆ0.750
1002
x
pn
== ≈ 4. 2439
ˆ0.830
2939
x
pn
== ≈
ˆˆ
1 0.250qp=− ≈
ˆˆ
1 0.170qp=− ≈
9. (0.512, 0.596) ˆ0.554 0.596 0.554 0.042pE→= ⇒= − =
10. (0.087, 0.263) ˆ0.175 0.263 0.175 0.088pE→= ⇒= − =
12. 279
ˆ0.664
420
x
pn
== ≈
ˆˆ
1 0.336qp=− ≈
90% CI: ˆˆ (0.664)(0.336)
ˆ0.664 1.645 0.664 0.0379
420
(0.626, 0.702)
c
pq
pz n
±≈± ≈±
=