240 CHAPTER 6 │ CONFIDENCE INTERVALS
With 90% confidence, you can say that the population proportion of U.S. females ages 18-64
who say they have gone to the dentist in the past year is between 62.6% and 70.2%. With 95%
confidence, you can say it is between 61.9% and 70.9%. The 95% confidence interval is slightly
wider.
14. 722
ˆ0.180
4013
x
pn
== ≈
ˆˆ
1 0.820qp=− ≈
ˆˆ (0.180)(0.820)
ˆ0.180 2.575 0.180 0.016
4013
(0.164, 0.196)
c
pq
pz n
±≈± ≈±
=
With 99% confidence, you can say that the population proportion of U.S. adults who say they
have seen a ghost is between 16.4% and 19.6%.
16. 184
ˆ0.207
891
x
pn
== ≈
ˆˆ
1 0.793qp=− ≈
CHAPTER 6 │ CONFIDENCE INTERVALS 241
18. (a)
22
2.575
ˆˆ 0.5 0.5 4144.14 4145 adults
0.02
c
z
npq
E
⎛⎞ ⎛⎞
==⋅ ≈→
⎜⎟
⎜⎟ ⎝⎠
⎝⎠
(b)
22
2.575
ˆˆ 0.47 0.53 4129.22 4130 adults
0.02
c
z
npq
E
⎛⎞ ⎛⎞
==⋅ ≈→
⎜⎟
⎜⎟ ⎝⎠
⎝⎠
(c) Having an estimate of the population proportion reduces the minimum sample size needed.
20. (a)
22
1.96
ˆˆ 0.5 0.5 384.16 385 adults
0.05
c
z
npq
E
⎛⎞ ⎛⎞
==⋅ ≈→
⎜⎟
⎜⎟ ⎝⎠
⎝⎠
(b)
22
1.96
ˆˆ 0.27 0.73 302.87 303 adults
0.05
c
z
npq
E
⎛⎞ ⎛⎞
==⋅ ≈→
⎜⎟
⎜⎟ ⎝⎠
⎝⎠
(c) Having an estimate of the population proportion reduces the minimum sample size needed.
242 CHAPTER 6 │ CONFIDENCE INTERVALS
(c)
ˆˆ
0.31, 0.69, 1126pqn===
ˆˆ (0.31)(0.69)
ˆ0.31 2.575 0.31 0.035 (0.275, 0.345)
1126
c
pq
pz n
±=± ≈±=
23. (a) ˆ0.32p= ˆˆ
10.68qp=− =
ˆˆ (0.32)(0.68)
ˆ0.32 1.96 0.32 0.046
400
(0.274, 0.366)
c
pq
pz n
±=± ≈±
=
(b)
ˆ0.56p= ˆˆ
10.44qp=− =
ˆˆ (0.56)(0.44)
ˆ0.56 1.96 0.56 0.049
400
(0.511, 0.609)
c
pq
pz n
±=± ≈±
=
CHAPTER 6 │ CONFIDENCE INTERVALS 243
26. No, it is unlikely that the two proportions are equal because the confidence intervals estimating
the proportions do not overlap. The 99% confidence intervals are (0.298, 0.422) and (0.204,
0.316). Using these intervals, it is possible that the two proportions are equal because the
confidence intervals overlap.
With 90% confidence, you can say the population proportion of U.S. adults who disapprove of
the job Congress is doing is between 76.1% and 80.4%. With 95% confidence you can say it is
between 75.7% and 80.8%. With 99% confidence you can say it is between 74.9% and 81.6%.
28. 90% confidence interval results:
p: proportion of successes for population
method: Standard-Wald
Proportion Count Total Sample
Prop.
Std. err. L. Limit U Limit
p 734 2303 0.3187147 0.009709986 0.30274323 0.33468622
244 CHAPTER 6 │ CONFIDENCE INTERVALS
99% confidence interval results:
p: proportion of successes for population
method: Standard-Wald
29. 31.4% 1% (30.4%, 32.4%) (0.304, 0.324)±→ →
ˆˆ 8451
0.01 1.981 1.98 0.9761
ˆˆ 0.314 0.686
cc c
pq n
Ez z E z c
npq
=→== ≈→=→=
−
(30.4%, 32.4%) is approximately a 95.23% CI.
30. 19% 3% (16%, 22%) (0.16, 0.22)±→ →
32. Sample answer:
ˆˆ
c
pq
Ez n
=
Write original equation.
CHAPTER 6 │ CONFIDENCE INTERVALS 245
33.
ˆ
p ˆˆ
1qp=− ˆˆ
pq ˆ
p ˆˆ
1qp=− ˆˆ
pq
0.0
0.1
0.2
0.3
1.0
0.9
0.8
0.7
0.00
0.09
0.16
0.21
0.45
0.46
0.47
0.48
0.55
0.54
0.53
0.52
0.2475
0.2484
0.2491
0.2496
6.4 CONFIDENCE INTERVALS FOR VARIANCE AND STANDARD
DEVIATION
6.3 Try It Yourself Solutions
1a. d.f. = n − 1 = 29
level of confidence = 0.90
χ
χ
χ
2a. 90% CI: 2
R
χ
= 42.557, 2
L
χ
= 17.708
95% CI: 2
R
χ
= 42.722, 2
L
χ
= 16.047
b. 90% CI for
22 2 2
2
22
( 1) ( 1) 29 (1.2) 29 (1.2)
: , (0.98, 2.36)
42.557 17.708
RL
nsns
σ
χχ
⎛⎞
⎛⎞
−− ⋅ ⋅
=⋅≈
⎜⎟
⎜⎟
⎝⎠
⎝⎠
246 CHAPTER 6 │ CONFIDENCE INTERVALS
6.4 EXERCISE SOLUTIONS
1. Yes.
2. It approaches the shape of the normal curve.
9. (a) s ≈ 0.00843
22 2 2
22
( 1) ( 1) 13 (0.00843) 13 (0.00843) (0.0000413, 0.000157)
22.362 5.892
RL
nsns
χχ
⎛⎞
⎛⎞
−− ⋅ ⋅
⋅≈ ⋅ ≈
⎜⎟
⎜⎟
⎝⎠
⎝⎠
(b)
()
0.0000413, 0.000157 (0.00643, 0.0125)≈
With 90% confidence, you can say that the population variance is between 0.0000413 and
0.000157, and the population standard deviation is between 0.00643 and 0.0125 milligrams.
11. (a) s ≈ 0.253
22 2 2
22
( 1) ( 1) 17 (0.253) 17 (0.253) (0.0305, 0.191)
35.718 5.697
RL
nsns
χχ
⎛⎞
⎛⎞
−− ⋅ ⋅
⋅≈ ⋅ ≈
⎜⎟
⎜⎟
⎝⎠
⎝⎠
(b)
()
0.0305, 0.191 (0.175, 0.437)≈
With 99% confidence, you can say that the population variance is between 0.0305 and 0.191,
and the population standard deviation is between 0.175 and 0.437 hour.
CHAPTER 6 │ CONFIDENCE INTERVALS 247
(b)
()
0.00467, 0.0195 (0.0683, 0.140)≈
With 95% confidence, you can say that the population variance is between 0.00467 and
0.0195, and the population standard deviation is between 0.0683 and 0.0140 inch.
14. (a)
22 2 2
22
( 1) ( 1) 10 (109) 10 (109) (7432, 24,421)
15.987 4.865
RL
nsns
χχ
⎛⎞
⎛⎞
−− ⋅ ⋅
⋅≈ ⋅ ≈
⎜⎟
⎜⎟
⎝⎠
⎝⎠
(b)
()
7432, 24,421 (86, 156)≈
With 80% confidence, you can say that the population variance is between 7432 and 24.421,
and the population standard deviation is between $86 and $156.
16. (a)
22 2 2
22
( 1) ( 1) 13 (0.875) 13 (0.875) (0.33, 2.79)
29.819 3.565
RL
nsns
χχ
⎛⎞
⎛⎞
−− ⋅ ⋅
⋅≈ ⋅ ≈
⎜⎟
⎜⎟
⎝⎠
⎝⎠
(b)
()
0.334, 2.793 (0.58, 1.67)≈
With 99% confidence, you can say that the population variance is between 0.33 and 2.79, and
the population standard deviation is between 0.58 and 1.67 pounds.
248 CHAPTER 6 │ CONFIDENCE INTERVALS
18. (a)
22 2 2
22
( 1) ( 1) 26 (138.33) 26 (138.33) (10,901.37, 40,789.60)
45.642 12.198
RL
nsns
χχ
⎛⎞
⎛⎞
−− ⋅ ⋅
⋅≈ ⋅ ≈
⎜⎟
⎜⎟
⎝⎠
⎝⎠
(b)
()
10,901.37, 40,789.60 (104, 202)≈
With 98% confidence, you can say that the population variance is between 10,901.06 and
40,789.17, and the population standard deviation is between $104 and $202.
20. (a)
22 2 2
22
( 1) ( 1) 29(3600) 29(3600) (8,831, 450, 21,224,305)
42.557 17.708
RL
nsns
χχ
⎛⎞
⎛⎞
−−
⋅≈ ⋅ ≈
⎜⎟
⎜⎟
⎝⎠
⎝⎠
(b)
()
8,831,450, 21,224,305 (2972, 4607)≈
With 90% confidence, you can say that the population variance is between 8,831,450 and
21,224,305, and the population standard deviation is between $2972 and $4607.
22. (a)
22 2 2
22
( 1) ( 1) 29 (8.18) 29 (8.18) (39.13, 136.11)
49.588 14.256
RL
nsns
χχ
⎛⎞
⎛⎞
−− ⋅ ⋅
⋅≈ ⋅ ≈
⎜⎟
⎜⎟
⎝⎠
⎝⎠
CHAPTER 6 │ CONFIDENCE INTERVALS 249
23. (a)
22 2 2
22
( 1) ( 1) (21)(3.6) (21)(3.6) (7.0, 30.6)
38.932 8.897
RL
nsns
χχ
⎛⎞
⎛⎞
−−
⋅≈ ⋅ ≈
⎜⎟
⎜⎟
⎝⎠
⎝⎠
24. (a)
22 2 2
22
( 1) ( 1) 19(3900) 19(3900) (9,586,982, 28,564,792)
30.144 10.117
RL
nsns
χχ
⎛⎞
⎛⎞
−−
⋅≈ ⋅ ≈
⎜⎟
⎜⎟
⎝⎠
⎝⎠
25. 95% confidence interval results:
2:
σ
variance of variable
Variance Sample Var. DF L. Limit U Limit
2
σ
11.56 29 7.332092 20.891039
(2.71, 4.57)
27. 90% confidence interval results:
2:
σ
variance of variable
28. 99% confidence interval results:
2:
σ
variance of variable
Variance Sample Var. DF L. Limit U Limit
2
σ
77339.6 44 50970.645 130062.46
(225.8, 360.6)
250 CHAPTER 6 │ CONFIDENCE INTERVALS
29. 90% CI for : (0.00643, 0.0125)
σ
Yes, because all of the values in the confidence interval are less than 0.015.
CHAPTER 6 REVIEW EXERCISE SOLUTIONS
1. (a) 103.5x≈
(b) 34.663s≈
34.663
1.645 9.0
40
c
s
Ez n
=≈ ≈
4. 0.105
7.675 1.96 7.675 0.028 (7.647, 7.703)
55
c
s
xz n
± =± ≈±≈
5. (20.75, 24.10) 22.425 24.10 22.425 1.675xE→= →= − =
6. (7.428, 7.562) 7.495 7.562 7.495 0.067xE→= →= − =
9.
22
(1.96)(7.098) 48.39 49 people
2
c
z
nE
σ
⎛⎞
⎛⎞
=≈ ≈⇒
⎜⎟
⎜⎟
⎝⎠
⎝⎠
CHAPTER 6 │ CONFIDENCE INTERVALS 251
17. 25.6
1.753 11.2
16
c
s
Et n
== ≈ 18. 1.1
2.064 0.5
25
c
s
Et n
== ≈
19. 0.9
2.718 0.7
12
c
s
Et n
⎛⎞
== ≈
⎜⎟
⎝⎠ 20. 16.5
2.861 10.6
20
c
s
Et n
⎛⎞
== ≈
⎜⎟
⎝⎠
21. 25.6
72.1 1.753 72.1 11.219 (60.9, 83.3)
16
c
s
xz n
±=± ≈± ≈
25. 523
2218 1.703 2218 168.3 (2050, 2386)
28
c
s
xt n
±=± ≈±≈
26. 523
2218 2.771 2218 273.9 (1944, 2492)
28
c
s
xt n
±=± ≈±≈
27. 1215
ˆˆ
0.81, 0.19
1500
x
pq
n
== = = 28. 425
ˆˆ
0.85, 0.15
500
x
pq
n
== = =
252 CHAPTER 6 │ CONFIDENCE INTERVALS
33. 346
ˆˆ
0.490, 0.510
706
x
pq
n
== ≈ ≈ 34. 1230
ˆˆ
0.520, 0.480
2365
x
pq
n
== ≈ ≈
36. ˆˆ 0.85 0.15
ˆ0.85 2.575 0.85 0.041 (0.809, 0.891)
500
c
pq
pz n
⋅
±=± ≈±=
With 99% confidence, you can say that the population proportion of U.S. adults who say they
would trust doctors to tell the truth is between 80.9% and 89.1%.
38. ˆˆ 0.113 0.887
ˆ0.113 2.326 0.113 0.026 (0.087, 0.139)
800
c
pq
pz n
⋅
±≈± ≈±=
With 98% confidence, you can say that the population proportion of U.S. adults who say they are
making the minimum payment(s) on their credit card(s) is between 8.7% and 13.9%.
40. ˆˆ 0.251 0.749
ˆ0.251 1.645 0.251 0.023 (0.228, 0.274)
938
c
pq
pz n
⋅
±≈± ≈±=
With 90% confidence, you can say that the population proportion of U.S. adults who say the
phrase “you know” is the most annoying conversational phrase is between 22.8% and 27.4%.
CHAPTER 6 │ CONFIDENCE INTERVALS 253
worry most about missing deductions when filing their taxes is between 49.6% and 54.4%.
43. (a)
22
1.96
ˆˆ 0.50 0.50 384.16 385 adults
0.05
c
z
npq
E
⎛⎞ ⎛⎞
==⋅ ≈→
⎜⎟
⎜⎟ ⎝⎠
⎝⎠
(b)
22
1.96
ˆˆ 0.63 0.37 358.19 359 adults
0.05
c
z
npq
E
⎛⎞ ⎛⎞
==⋅ ≈→
⎜⎟
⎜⎟ ⎝⎠
⎝⎠
(c) The minimum sample size needed is smaller when a preliminary estimate is available.
χ
χ
χ
χ
47. 2
R
χ
= 14.067, 2
L
χ
= 2.167 48. 2
R
χ
= 23.589, 2
L
χ
= 1.735
49. 7.002s≈
50. 99% CI for
22 2 2
2
22
( 1) ( 1) 16 (7.002) 16 (7.002)
: , , (22.9, 152.6)
34.267 5.142
RL
ns ns
σ
χχ
⎛⎞
⎛⎞
−− ⋅ ⋅
≈≈
⎜⎟
⎜⎟
⎝⎠
⎝⎠
51. 1.190s≈
98% CI for
22 2
2
22
( 1) ( 1) (25)(1.190) (25)(1.190)
: , , (0.80, 3.07)
44.314 11.524
RL
ns ns
σ
χχ
⎛⎞
⎛⎞
−−
≈≈
⎜⎟
⎜⎟
⎝⎠
⎝⎠
98% CI for
()
: 0.80, 3.07 (0.89, 1.75)
σ
≈
254 CHAPTER 6 │ CONFIDENCE INTERVALS
52. 1.190s≈
98% CI for
22 2
2
22
( 1) ( 1) (25)(1.190) (25)(1.190)
: , , (0.94, 2.42)
37.652 14.611
RL
ns ns
σ
χχ
⎛⎞
⎛⎞
−−
≈≈
⎜⎟
⎜⎟
⎝⎠
⎝⎠
CHAPTER 6 QUIZ SOLUTIONS
1. (a) 6.85x≈
(b) 1.821s≈
1.821
1.960 0.65
30
c
s
Et n
=≈ ≈
2.
22
2.575 2.4 38.18 39 students
1
c
z
nE
σ
⋅
⎛⎞
⎛⎞
== ≈→
⎜⎟
⎜⎟
⎝⎠
⎝⎠
3. (a) 33.1x≈
2.38s≈
CHAPTER 6 │ CONFIDENCE INTERVALS 255
5. (a) 1079
ˆ0.780
1383
x
pn
== ≈
(b) ˆˆ 0.780 0.220
ˆ0.780 1.645 0.780 0.018 (0.762, 0.798)
1383
c
pq
pz n
⋅
±≈± ≈±=