1. No category

Hypothesis Testing with One Sample
CHAPTER
7
7.1 INTRODUCTION TO HYPOTHESIS TESTING
7.1 Try It Yourself Solutions
1a. (1) The mean . . . is not 74 months.
␮ ⫽ 74
(2) The variance . . . is less than or equal to 3.5.
␴ 2 ⱕ 3.5
(3) The proportion . . . is greater than 39%.
p > 0.39
b. (1) ␮ ⫽ 74
(2) ␴ 2 > 3.5
(3) p ⱕ 0.39
c. (1) H0 : ␮ ⫽ 74; Ha : ␮ ⫽ 74; (claim)
(2) H0 : ␴ 2 ⱕ 3.5 (claim); Ha : ␴ 2 > 3.5
(3) H0 : p ⱕ 0.39; Ha : p > 0.39 (claim)
2a. H0 : p ⱕ 0.01; Ha : p > 0.01
b. Type I error will occur if the actual proportion is less than or equal to 0.01, but you
reject H0 .
Type II error will occur if the actual proportion is greater than 0.01, but you fail to
reject H0 .
c. Type II error is more serious because you would be misleading the consumer, possibly
causing serious injury or death.
3a. (1) H0 : ␮ ⫽ 74; Ha : ␮ ⫽ 74
(2) H0 : p ⱕ 0.39; Ha : p > 0.39
b. (1) Two-tailed
c. (1)
(2) Right-tailed
1
P-value
2
area
−z
1
P-value
2
area
(2)
z
z
P-value
area
z
z
4a. There is enough evidence to support the radio station’s claim.
b. There is not enough evidence to support the radio station’s claim.
5a. (1) Support claim.
(2) Reject claim.
b. (1) H0: ␮ ⱖ 650; Ha: ␮ < 650 (claim)
(2) H0: ␮ ⫽ 98.6 (claim); Ha: ␮ ⫽ 98.6
179
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7.1 EXERCISE SOLUTIONS
1. Null hypothesis 共H0兲 and alternative hypothesis 共Ha兲. One represents the claim, the other, its
complement.
2. Type I Error: The null hypothesis is rejected when it is true.
Type II Error: The null hypothesis is not rejected when it is false.
3. False. In a hypothesis test, you assume the null hypothesis is true.
4. False. A statistical hypothesis is a statement about a population.
5. True
6. True
7. False. A small P-value in a test will favor a rejection of the null hypothesis.
8. False. If you want to support a claim, write it as your alternative hypothesis.
9. H0 : ␮ ⱕ 645 (claim); Ha : ␮ > 645
10. H0 : ␮ ⱖ 128; Ha : ␮ < 128 (claim)
11. H0 : ␴ ⫽ 5; Ha : ␴ ⫽ 5 (claim)
12. H0 : ␴ 2 ⱖ 1.2 (claim); Ha : ␴ 2 < 1.2
13. H0 : p ⱖ 0.45; Ha: p < 0.45 (claim)
14. H0 : p ⫽ 0.21; Ha : p ⫽ 0.21
H0 : ␮ ⱕ 3
15. c,
16. d, Ha : ␮ ⱖ 3
μ
1
2
3
μ
4
1
H0 : ␮ ⫽ 3
17. b,
2
3
4
18. a, Ha: ␮ ⱕ 2
μ
1
2
3
19. Right-tailed
μ
4
1
20. Left-tailed
2
3
21. Two-tailed
4
22. Two-tailed
23. ␮ > 750
H0: ␮ ⱕ 750; Ha: ␮ > 750 (claim)
24. ␴ < 3
H0 : ␴ ⱖ 3; Ha : ␴ < 3 (claim)
25. ␴ ⱕ 320
H0: ␴ ⱕ 320 (claim); Ha: ␴ > 320
26. p ⫽ 0.28
H0 : p ⫽ 0.28 (claim); Ha : p ⫽ 0.28
27. p ⫽ 0.81
H0 : p ⫽ 0.81 (claim); Ha: p ⫽ 0.81
28. ␮ < 45
H0 : ␮ ⱖ 45; Ha : ␮ < 45 (claim)
29. Type I: Rejecting H0: p ⱖ 0.60 when actually p ⱖ 0.60.
Type II: Not rejecting H0: p ⱖ 0.60 when actually p < 0.60.
30. Type I: Rejecting H0 : p ⫽ 0.05 when actually p ⫽ 0.05.
Type II: Not rejecting H0 : p ⫽ 0.05 when actually p ⫽ 0.05.
31. Type I: Rejecting H0: ␴ ⱕ 12 when actually ␴ ⱕ 12.
Type II: Not rejecting H0: ␴ ⱕ 12 when actually ␴ > 12.
32. Type I: Rejecting H0 : p ⫽ 0.50 when actually p ⫽ 0.50.
Type II: Not rejecting H0 : p ⫽ 0.50 when actually p ⫽ 0.50.
33. Type I: Rejecting H0: p ⫽ 0.88 when actually p ⫽ 0.88.
Type II: Not rejecting H0: p ⫽ 0.88 when actually p ⫽ 0.88.
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CHAPTER 7
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HYPOTHESIS TESTING WITH ONE SAMPLE
34. Type I: Rejecting p ⫽ 0.30 when actually p ⫽ 0.30.
Type II: Not rejecting H0 : p ⫽ 0.30 when actually p ⫽ 0.30.
35. The null hypothesis is H0: p ⱖ 0.14, the alternative hypothesis is Ha: p < 0.14.
Therefore, because the alternative hypothesis contains <, the test is a left-tailed test.
36. The null hypothesis is H0 : ␮ ⱕ 0.02, the alternative hypothesis is Ha: ␮ > 0.02.
Therefore, because the alternative hypothesis contains >, the test is a right-tailed test.
37. The null hypothesis is H0: p ⫽ 0.87, the alternative hypothesis is Ha: p ⫽ 0.87.
Therefore, because the alternative hypothesis contains ≠, the test is a two-tailed test.
38. The null hypothesis is H0 : ␮ ⱖ 80,000, the alternative hypothesis is Ha: ␮ < 80,000.
Therefore, because the alternative hypothesis contains <, the test is a left-tailed test.
39. The null hypothesis is H0: p ⫽ 0.053, the alternative hypothesis is Ha: p ⫽ 0.053.
Therefore, because the alternative hypothesis contains ≠, the test is a two-tailed test.
40. The null hypothesis is H0 : ␮ ⱕ 10, the alternative hypothesis is Ha: ␮ > 10.
Therefore, because the alternative hypothesis contains >, the test is a right-tailed test.
41. (a) There is enough evidence to support the company’s claim.
(b) There is not enough evidence to support the company’s claim.
42. (a) There is enough evidence to reject the government worker’s claim.
(b) There is not enough evidence to reject the government worker’s claim.
43. (a) There is enough evidence to support the Department of Labor’s claim.
(b) There is not enough evidence to support the Department of Labor’s claim.
44. (a) There is enough evidence to reject the manufacturer’s claim.
(b) There is not enough evidence to reject the manufacturer’s claim.
45. (a) There is enough evidence to support the manufacturer’s claim.
(b) There is not enough evidence to support the manufacturer’s claim.
46. (a) There is enough evidence to reject the soft-drink maker’s claim.
(b) There is not enough evidence to reject the soft-drink maker’s claim.
47. H0 : ␮ ⱖ 60; Ha: ␮ < 60
48. H0: ␮ ⫽ 21; Ha: ␮ ⫽ 21
49. (a) H0: ␮ ⱖ 15; Ha: ␮ < 15
50. (a) H0 : ␮ ⱕ 28; Ha : ␮ > 28
(b) H0: ␮ ⱕ 15 Ha: ␮ > 15
(b) H0 : ␮ ⱖ 28; Ha : ␮ < 28
51. If you decrease ␣, you are decreasing the probability that you reject H0. Therefore, you are
increasing the probability of failing to reject H0. This could increase ␤, the probability of
failing to reject H0 when H0 is false.
52. If ␣ ⫽ 0, the null hypothesis cannot be rejected and the hypothesis test is useless.
53. (a) Fail to reject H0 because the CI includes values greater than 70.
(b) Reject H0 because the CI is located below 70.
(c) Fail to reject H0 because the CI includes values greater than 70.
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54. (a) Fail to reject H0 because the CI includes values less than 54.
(b) Fail to reject H0 because the CI includes values less than 54.
(c) Reject H0 because the CI is located to the right of 54.
55. (a) Reject H0 because the CI is located to the right of 0.20.
(b) Fail to reject H0 because the CI includes values less than 0.20.
(c) Fail to reject H0 because the CI includes values less than 0.20.
56. (a) Fail to reject H0 because the CI includes values greater than 0.73.
(b) Reject H0 because the CI is located to the left of 0.73.
(c) Fail to reject H0 because the CI includes values greater than 0.73.
7.2 HYPOTHESIS TESTING FOR THE MEAN
(LARGE SAMPLES)
7.2 Try It Yourself Solutions
1a. (1) P ⫽ 0.0347 > 0.01 ⫽ ␣
(2) P ⫽ 0.0347 < 0.05 ⫽ ␣
b. (1) Fail to reject H0 because 0.0347 > 0.01.
(2) Reject H0 because 0.0347 < 0.05.
2a.
Area =
0.0526
−3 −2 −1
z
0
z = − 1.62
1
2
3
b. P ⫽ 0.0526
c. Fail to reject H0 because P ⫽ 0.0526 > 0.05 ⫽ ␣.
3a. Area that corresponds to z ⫽ 2.31 is 0.9896.
Area =
0.0104
−3 −2 −1
z
0
1
2
3
z = 2.31
b. P ⫽ 2 共area兲 ⫽ 2共0.0104兲 ⫽ 0.0208
c. Fail to reject H0 because P ⫽ 0.0208 > 0.01 ⫽ ␣.
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CHAPTER 7
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HYPOTHESIS TESTING WITH ONE SAMPLE
4a. The claim is “the mean speed is greater than 35 miles per hour.”
H0 : ␮ ⱕ 35; Ha: ␮ > 35 (claim)
b. ␣ ⫽ 0.05
x ⫺ ␮ 36 ⫺ 35
1
⫽
⫽
⫽ 2.500
s
4
0.4
冪n
冪100
d. P-value ⫽ Area right of z ⫽ 2.50 ⫽ 0.0062
c. z ⫽
e. Reject H0 because P-value ⫽ 0.0062 < 0.05 ⫽ ␣.
f. Because you reject H0 , there is enough evidence to claim the average speed limit is greater
than 35 miles per hour.
5a. The claim is “one of your distributors reports an average of 150 sales per day.”
H0 : ␮ ⫽ 150 (claim); Ha : ␮ ⫽ 150
b. ␣ ⫽ 0.01
x ⫺ ␮ 143 ⫺ 150
⫺7
⫽
⫽
⬇ ⫺2.76
␴
15
2.535
冪n
冪35
d. P-value ⫽ 0.0058
c. z ⫽
e. Reject H0 because P-value ⫽ 0.0058 < 0.01 ⫽ ␣.
f. There is enough evidence to reject the claim.
6a. P ⫽ 0.0440 > 0.01 ⫽ ␣
b. Fail to reject H0.
7a.
8a.
1
α
2
α = 0.10
−3 −2 z 0
z
0
1
2
−3
3
1
α
2
= 0.04
−z 0
0
1 z0
= 0.04
z
3
b. Area ⫽ 0.1003
b. 0.0401 and 0.9599
c. z0 ⫽ ⫺1.28
c. z0 ⫽ ⫺1.75 and 1.75
d. z < ⫺1.28
d. z < ⫺1.75, z > 1.75
9a. The claim is “the mean work day of the firm’s accountants is less than 8.5 hours.”
H0: ␮ > 8.5; Ha: ␮ < 8.5 (claim)
b. ␣ ⫽ 0.01
c. z0 ⫽ ⫺2.33; Rejection region: z < ⫺2.33
d. z ⫽
x ⫺ ␮ 8.2 ⫺ 8.5 ⫺0.300
⫽
⫽
⬇ ⫺3.550
s
0.5
0.0845
冪n
冪35
Reject H0.
e.
α = 0.01
−3
z0
−1
z = − 3.55
z
0
1
2
3
f. There is enough evidence to support the claim.
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10a. ␣ ⫽ 0.01
b. ± z0 ⫽ ± 2.575; Rejection regions: z < ⫺2.575, z > 2.575
c. Fail to reject H0.
1
α
2
1
α
2
= 0.005
= 0.005
z = −2.24
− z0
−1
0
1
2
z
z0
d. There is not enough evidence to support the claim that the mean cost is significantly
different from $10,460 at the 1% level of significance.
7.2 EXERCISE SOLUTIONS
1.
2.
Area =
0.1151
−3 −2 −1
Area =
0.0455
z
0
z = −1.20
1
2
−3 −2 −1
3
z
0
z = −1.69
P ⫽ 0.1151; Fail to reject H0
because P ⫽ 0.1151 > 0.10 ⫽ ␣.
1
2
3
P ⫽ 0.0455; Reject H0
because P ⫽ 0.0455 < 0.05 ⫽ ␣.
3.
4.
Area =
0.0096
−3 −2 −1
Area =
0.1093
z
0
1
2
−3 −2 −1
3
z = 2.34
z
0
1
2
3
z = 1.23
P ⫽ 0.0096; Reject H0 because
P ⫽ 0.0096 < 0.01 ⫽ ␣.
P ⫽ 0.1093; Fail to reject H0
because P ⫽ 0.1093 > 0.10 ⫽ ␣.
5.
6.
Area =
0.0594
−3 −2 −1
Area =
0.0107
z
0
z = − 1.56
1
2
−3 −2 −1
3
8. d
1
2
3
z = 2.30
P ⫽ 2共Area兲 ⫽ 2共0.0594兲 ⫽ 0.1188;
Fail to reject H0 because P ⫽ 0.1188 > 0.05 ⫽ ␣.
7. c
z
0
9. e
10. f
11. b
P ⫽ 2共Area兲 ⫽ 2共0.0107兲 ⫽ 0.0214;
Fail to reject H0 because P ⫽ 0.0214 > 0.01 ⫽ ␣.
12. a
13. (a) Fail to reject H0 .
(b) Reject H0 共P ⫽ 0.0461 < 0.05 ⫽ ␣兲.
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CHAPTER 7
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HYPOTHESIS TESTING WITH ONE SAMPLE
14. (a) Fail to reject H0 共P ⫽ 0.0691 > 0.01 ⫽ ␣兲.
(b) Fail to reject H0 共P ⫽ 0.0691 > 0.05 ⫽ ␣兲.
15. 1.645
16. 1.41
17. ⫺1.88
18. ⫺1.34
19. ± 2.33
20. ± 1.645
21. Right-tailed 共␣ ⫽ 0.01兲
22. Two-tailed 共␣ ⫽ 0.05兲
23. Two-tailed 共␣ ⫽ 0.10兲
24. Left-tailed 共␣ ⫽ 0.05兲
25. (a) Fail to reject H0 because ⫺1.645 < z < 1.645.
(b) Reject H0 because z > 1.645.
(c) Fail to reject H0 because ⫺1.645 < z < 1.645.
(d) Reject H0 because z < ⫺1.645.
26. (a) Reject H0 because z > 1.96.
(b) Fail to reject H0 because ⫺1.96 < z < 1.96.
(c) Fail to reject H0 because ⫺1.96 < z < 1.96.
(d) Reject H0 because z < ⫺1.96.
27. (a) Fail to reject H0 because z < 1.285.
(b) Fail to reject H0 because z < 1.285.
(c) Fail to reject H0 because z < 1.285.
(d) Reject H0 because z > 1.285.
28. (a) Fail to reject H0 because ⫺2.575 < z < 2.575.
(b) Reject H0 because z < ⫺2.575.
(c) Reject H0 because z > 2.575.
(d) Fail to reject H0 because ⫺2.575 < z < 2.575.
29. H0 : ␮ ⫽ 40; Ha : ␮ ⫽ 40
␮ ⫽ 0.05 → z0 ⫽ ± 1.96
z⫽
x ⫺ ␮ 39.2 ⫺ 40
⫺0.8
⬇ ⫺2.145
⫽
⫽
s
3.23
0.373
冪n
冪75
Reject H0. There is enough evidence to reject the claim.
30. H0 : ␮ ⱕ 1030 and Ha : ␮ > 1030
␣ ⫽ 0.05 → z0 ⫽ 1.645
z⫽
x ⫺ ␮ 1035 ⫺ 1030
5
⫽
⫽
⬇ 1.537
s
23
3.253
冪n
冪50
Fail to reject H0. There is not enough evidence to support the claim.
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31. H0 : ␮ ⫽ 6000; Ha : ␮ ⫽ 6000
␣ ⫽ 0.01 → z0 ⫽ ± 2.575
z⫽
x ⫺ ␮ 5800 ⫺ 6000
⫺200
⫽
⫽
⬇ ⫺3.381
s
350
59.161
冪n
冪35
Reject H0. There is enough evidence to support the claim.
32. H0 : ␮ ⱕ 22,500 and Ha : ␮ > 22,500
␣ ⫽ 0.01 → z0 ⫽ 2.33
z⫽
x ⫺ ␮ 23,250 ⫺ 22,500
750
⫽
⫽
⬇ 4.193
s
1200
178.885
冪n
冪45
Reject H0 . There is enough evidence to reject the claim.
33. (a) H0 : ␮ ⱕ 275; Ha : ␮ > 275 (claim)
(b) z ⫽
x ⫺ ␮ 282 ⫺ 275
7
⫽
⫽
⬇ 1.84
s
35
3796
冪n
冪85
Area ⫽ 0.9671
(c) P-value ⫽ 再Area to right of z ⫽ 1.84冎 ⫽ 0.0329
(d) Reject H0 . There is sufficient evidence at the 4% level of significance to support the
claim that the mean score for Illinois’ eighth grades is more than 275.
34. (a) H0 : ␮ ⱖ 135 (claim); Ha : ␮ < 135
(b) z ⫽
x ⫺ ␮ 133 ⫺ 135
⫺2
⫽
⫽
⬇ ⫺3.43
s
3.3
0.583
冪n
冪32
Area ⫽ 0.0003
(c) P-value ⫽ 再Area to left of z ⫽ ⫺3.43冎 ⫽ 0.0003
(d) Reject H0 .
(e) There is sufficient evidence at the 10% level of significance to reject the claim that the
average activating temperature is at least 135⬚F.
35. (a) H0 : ␮ ⱕ 8; Ha : ␮ > 8 (claim)
(b) z ⫽
x ⫺ ␮ 7.9 ⫺ 8
⫺0.1
⫽
⫽
⬇ ⫺0.37
s
2.67
0.267
冪n
冪100
Area ⫽ 0.3557
(c) P-value ⫽ 再Area to right of z ⫽ ⫺0.37冎 ⫽ 0.6443
(d) Fail to reject H0 .
(e) There is insufficient evidence at the 7% level of significance to support the claim that
the mean consumption of tea by a person in the United States is more than 8 gallons
per year.
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CHAPTER 7
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HYPOTHESIS TESTING WITH ONE SAMPLE
36. (a) H0 : ␮ ⫽ 3.1 (claim); Ha : ␮ ⫽ 3.1
(b) z ⫽
x ⫺ ␮ 2.9 ⫺ 3.1
⫺0.2
⫽
⫽
⬇ ⫺1.65
s
0.94
0.121
冪n
冪60
Area ⫽ 0.0495
(c) P-value ⫽ 2再Area to left of z ⫽ ⫺1.65冎 ⫽ 2再0.0495冎 ⫽ 0.099
(d) Fail to reject H0.
(e) There is insufficient evidence at the 8% level to reject the claim that the mean tuna consumed by a person in the United States is 3.1 pounds per year.
37. (a) H0: ␮ ⫽ 15 (claim); Ha : ␮ ⫽ 15
(b) x ⬇ 14.834
z⫽
s ⬇ 4.288
x ⫺ ␮ 14.834 ⫺ 15 ⫺0.166
⫽
⫽
⬇ ⫺0.219
s
4.288
0.758
冪n
冪32
Area ⫽ 0.4129
(c) P-value ⫽ 2再Area to left of z ⫽ ⫺0.22冎 ⫽ 2再0.4129冎 ⫽ 0.8258
(d) Fail to reject H0 .
(e) There is insufficient evidence at the 5% level of significance to reject the claim that the
mean time it takes smokers to quit smoking permanently is 15 years.
38. (a) H0 : ␮ ⱕ $100,800; Ha : ␮ > $100,800 (claim)
(b) x ⬇ 94,891.47,
z⫽
s ⬇ 5239.516
x ⫺ ␮ 94,891.47 ⫺ 100,800 ⫺5908.53
⫽
⫽
⬇ ⫺6.58
s
5239.516
898.570
冪n
冪34
Area ⬇ 0
(c) P-value ⫽ 再Area to right of z ⫽ ⫺6.58冎 ⬇ 1
(d) Fail to reject H0 .
(e) There is insufficient evidence at the 3% level to support the claim that the mean annual
salary for engineering managers in Alabama is at least $100,800.
39. (a) H0 : ␮ ⫽ 40 (claim); Ha : ␮ ⫽ 40
(b) z0 ⫽ ± 2.575;
Rejection regions: z < ⫺2.575 and z > 2.575
(c) z ⫽
x ⫺ ␮ 39.2 ⫺ 40
⫺0.8
⫽
⫽
⬇ ⫺0.584
s
7.5
1.369
冪n
冪30
(d) Fail to reject H0.
(e) There is insufficient evidence at the 1% level of significance to reject the claim that the
mean caffeine content per one 12-ounce bottle of cola is 40 milligrams.
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40. (a) H0 : ␮ ⫽ 140 (claim); Ha : ␮ ⫽ 140
(b) z0 ⫽ ± 1.96; Rejection regions: z < ⫺1.96 and z > 1.96
(c) z ⫽
x ⫺ ␮ 146 ⫺ 140
6
⫽
⫽
⬇ 1.77
s
22
3.395
冪n
冪42
(d) Fail to reject H0 .
(e) There is insufficient evidence at the 5% level to reject the claim that the mean caffeine
content is 140 milligrams per 8 ounces.
41. (a) H0 : ␮ ⱖ 750 (claim); Ha : ␮ < 750
(b) z0 ⫽ ⫺2.05; Rejection region: z < ⫺2.05
(c) z ⫽
x ⫺ ␮ 745 ⫺ 750 ⫺5
⫽
⫽
⬇ ⫺0.500
s
60
10
冪n
冪36
(d) Fail to reject H0 .
(e) There is insufficient evidence at the 2% level of significance to reject the claim that the
mean life of the bulb is at least 750 hours.
42. (a) H0 : ␮ ⱕ 230; Ha : ␮ > 230 (claim)
(b) z0 ⫽ 1.75; Rejection region: z > 1.75
(c) z ⫽
x ⫺ ␮ 232 ⫺ 230
2
⫽
⫽
⬇ 1.44
s
10
1.387
冪n
冪52
(d) Fail to reject H0 .
(e) There is insufficient evidence at the 4% level to support the claim that the mean sodium
content per serving of cereal is greater than 230 milligrams.
43. (a) H0 : ␮ ⱕ 32; Ha : ␮ > 32 (claim)
(b) z0 ⫽ 1.55; Rejection region: z > 1.55
(c) x ⬇ 29.676
z⫽
s ⬇ 9.164
x ⫺ ␮ 29.676 ⫺ 32 ⫺2.324
⫽
⫽
⬇ ⫺1.478
s
9.164
1.572
冪n
冪34
(d) Fail to reject H0 .
(e) There is insufficient evidence at the 6% level of significance to support the claim that the
mean nitrogen dioxide level in Calgary is greater than 32 parts per billion.
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HYPOTHESIS TESTING WITH ONE SAMPLE
44. (a) H0: ␮ ⱖ 10,000 (claim); Ha : ␮ < 10,000
(b) z0 ⫽ ⫺1.34; Rejection region: z < ⫺1.34
(c) x ⫽ 9580.9,
z⫽
s ⫽ 1722.4
x ⫺ ␮ 9580.9 ⫺ 10,000 ⫺419.1
⫽
⫽
⫽ ⫺1.38
s
1722.4
304.5
冪n
冪32
(d) Reject H0.
(e) There is sufficient evidence at the 9% level to reject the claim that the mean life of
fluorescent lamps is at least 10,000 hours.
45. (a) H0: ␮ ⱖ 10 (claim); Ha : ␮ < 10
(b) z0 ⫽ ⫺1.88; Rejection region: z < ⫺1.88
(c) x ⬇ 9.780, s ⬇ 2.362
x⫽
x ⫺ ␮ 9.780 ⫺ 10 ⫺0.22
⫽
⫽
⬇ ⫺0.51
s
2.362
0.431
冪n
冪30
(d) Fail to reject H0 .
(e) There is insufficient evidence at the 3% level to reject the claim that the mean weight loss
after 1 month is at least 10 pounds.
46. (a) H0: ␮ ⱖ 60; Ha : ␮ < 60 (claim)
(b) z0 ⫽ ⫺2.33; Rejection region: z < ⫺2.33
(c) x ⫽ 49,
z⫽
s ⫽ 21.51
x ⫺ ␮ 49 ⫺ 60
⫺11
⫽
⫽
⫽ ⫺3.62
s
21.51
3.042
冪n
冪50
(d) Fail to reject H0 .
(e) There is sufficient evidence at the 1% level to support the claim that the mean time it
takes an employee to evacuate a building during a fire drill is less than 60 seconds.
47. z ⫽
x ⫺ ␮ 11,400 ⫺ 11,500
⫺100
⬇ ⫺1.71
⫽
⫽
s
320
58.424
冪n
冪30
P-value ⫽ 再Area left of z ⫽ ⫺1.71冎 ⫽ 0.0436
Fail to reject H0 because the standardized test statistic z ⫽ ⫺1.71 is greater than the critical
value z0 ⫽ ⫺2.33.
48. z ⫽
x ⫺ ␮ 22,200 ⫺ 22,000
200
⫽
⫽
⫽ 1.548
s
775
129.2
冪n
冪36
P-value ⫽ 再Area right of z ⫽ 1.55冎 ⫽ 0.0606
Fail to reject H0 because P-value ⫽ 0.0606 > 0.05 ⫽ ␣.
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49. (a) ␣ ⫽ 0.02; Fail to reject H0.
(b) ␣ ⫽ 0.05; Reject H0.
(c) z ⫽
x ⫺ ␮ 11,400 ⫺ 11,500
⫺100
⫽
⫽
⬇ ⫺2.21
s
320
45.254
冪n
冪50
P-value ⫽ 再Area left of z ⫽ ⫺2.21冎 ⫽ 0.0136 → Fail to reject H0.
(d) z ⫽
x ⫺ ␮ 11,400 ⫺ 11,500 ⫺100
⫽
⫽
⬇ ⫺3.13
s
320
32
冪n
冪100
P-value ⫽ 再Area left of z ⫽ ⫺3.13冎 < 0.0009 → Reject H0.
50. (a) ␣ ⫽ 0.06; Fail to reject H0.
(b) ␣ ⫽ 0.07; Reject H0.
(c) z ⫽
x ⫺ ␮ 22,200 ⫺ 20,000
200
⫽
⫽
⫽ 1.63
s
775
122.5
冪n
冪40
P-value ⫽ 再Area right of z ⫽ 1.63冎 ⫽ 0.0516; Fail to reject H0.
(d) z ⫽
x ⫺ ␮ 22,200 ⫺ 20,000
200
⫽
⫽
⬇ 2.31
s
775
86.6
冪n
冪80
P-value ⫽ 再Area right of z ⫽ 2.31冎 ⫽ 0.0104; Reject H0.
51. Using the classical z-test, the test statistic is compared to critical values. The z-test using a
P-value compares the P-value to the level of significance ␣.
7.3 HYPOTHESIS TESTING FOR THE MEAN
(SMALL SAMPLES)
7.3 Try It Yourself Solutions
1a. 2.650
b. t0 ⫽ ⫺2.650
2a. 1.860
b. t0 ⫽ ⫹1.860
3a. 2.947
b. t0 ⫽ ± 2.947
4a. The claim is “the mean cost of insuring a 2005 Honda Pilot LX is at least $1350.”
H0 : ␮ ⱖ $1350 (claim); Ha : ␮ < $1350
b. ␣ ⫽ 0.01 and d.f. ⫽ n ⫺ 1 ⫽ 8
c. t0 ⫽ ⫺2.896; Reject H0 if t ⱕ ⫺2.896.
d. t ⫽
x ⫺ ␮ 1290 ⫺ 1350
⫺60
⫽
⫽
⬇ ⫺2.571
s
70
23.333
冪n
冪9
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HYPOTHESIS TESTING WITH ONE SAMPLE
e. Fail to reject H0 .
α = 0.01
t0
t = − 2.571
−1
0
t
1
2
3
f. There is not enough evidence to reject the claim.
5a. The claim is “the mean conductivity of the river is 1890 milligrams per liter.”
H0 : ␮ ⫽ 1890 (claim); Ha : ␮ ⫽ 1890
b. ␣ ⫽ 0.01 and d.f. ⫽ n ⫺ 1 ⫽ 18
c. t0 ⫽ ± 2.878; Reject H0 if t < ⫺2.878 or t > 2.878.
d. t ⫽
x ⫺ ␮ 2500 ⫺ 1890
610
⫽
⫽
⬇ 3.798
s
700
160.591
冪n
冪19
e. Reject H0.
1
α
2
= 0.005
− 4 − t0
1
α
2
= 0.005
t = 3.798
− 1 0 1 2 t0 4
t
f. There is enough evidence to reject the company’s claim.
6a. t ⫽
x ⫺ ␮ 172 ⫺ 185
⫺13
⫽
⫽
⬇ ⫺2.123
s
15
6.124
冪n
冪6
P-value ⫽ 再Area left of t ⫽ ⫺2.123冎 ⬇ 0.0436
b. P-value ⫽ 0.0436 < 0.05 ⫽ ␣
c. Reject H0 .
d. There is enough evidence to reject the claim.
7.3 EXERCISE SOLUTIONS
1. Identify the level of significance ␣ and the degrees of freedom, d.f. ⫽ n ⫺ 1. Find the critical
value(s) using the t-distribution table in the row with n ⫺ 1 d.f. If the hypothesis test is:
(1) Left-tailed, use “One Tail, ␣” column with a negative sign.
(2) Right-tailed, use “One Tail, ␣” column with a positive sign.
(3) Two-tailed, use “Two Tail, ␣” column with a negative and a positive sign.
2. Identify the claim. State H0 and Ha . Specify the level of significance. Identify the degrees of
freedom and sketch the sampling distribution. Determine the critical value(s) and rejection
region(s). Find the standardized test statistic. Make a decision and interpret it in the context
of the original claim. The population must be normal or nearly normal.
3. t0 ⫽ 1.717
4. t0 ⫽ 2.764
5. t0 ⫽ ⫺2.101
7. t0 ⫽ ± 2.779
8. t0 ⫽ ± 2.262
9. 1.328
11. ⫺2.473
12. ⫺3.106
13. ± 3.747
6. t0 ⫽ ⫺1.771
10. 1.895
14. ± 1.721
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15. (a) Fail to reject H0 because t > ⫺2.086.
(b) Fail to reject H0 because t > ⫺2.086.
(c) Fail to reject H0 because t > ⫺2.086.
(d) Reject H0 because t < ⫺2.086.
16. (a) Fail to reject H0 because ⫺1.372 < t < 1.372.
(b) Reject H0 because t < ⫺1.372.
(c) Reject H0 because t > 1.372.
(d) Fail to reject H0 because ⫺1.372 < t < 1.372.
17. (a) Fail to reject H0 because ⫺2.602 < t < 2.602.
(b) Fail to reject H0 because ⫺2.602 < t < 2.602.
(c) Reject H0 because t > 2.602.
(d) Reject H0 because t < ⫺2.602.
18. (a) Fail to reject H0 because ⫺1.725 < t < 1.725.
(b) Reject H0 because t < ⫺1.725.
(c) Fail to reject H0 because ⫺1.725 < t < 1.725.
(d) Reject H0 because t > 1.725.
19. H0 : ␮ ⫽ 15 (claim); Ha : ␮ ⫽ 15
␣ ⫽ 0.01 and d.f. ⫽ n ⫺ 1 ⫽ 5
t0 ⫽ ± 4.032
t⫽
x ⫺ ␮ 13.9 ⫺ 15
⫺1.1
⫽
⫽
⬇ ⫺0.834
s
3.23
1.319
冪n
冪6
Fail to reject H0. There is not enough evidence to reject the claim.
20. H0 : ␮ ⱕ 25; Ha : ␮ > 25 (claim)
␣ ⫽ 0.05 and d.f. ⫽ n ⫺ 1 ⫽ 16
t0 ⫽ 1.746
t⫽
x ⫺ ␮ 26.2 ⫺ 25
1.2
⬇ 2.133
⫽
⫽
s
2.32
0.563
冪n
冪17
Reject H0. There is enough evidence to support the claim.
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HYPOTHESIS TESTING WITH ONE SAMPLE
21. H0: ␮ ⱖ 8000 (claim); Ha: ␮ < 8000
␣ ⫽ 0.01 and d.f. ⫽ n ⫺ 1 ⫽ 24
t0 ⫽ ⫺2.492
t⫽
x ⫺ ␮ 7700 ⫺ 8000 ⫺300
⫽
⫽
⬇ ⫺3.333
s
450
90
冪n
冪25
Reject H0. There is enough evidence to reject the claim.
22. H0 : ␮ ⫽ 52,200; Ha : ␮ ⫽ 52,200 (claim)
␣ ⫽ 0.05 and d.f. ⫽ n ⫺ 1 ⫽ 3
t0 ⫽ ± 3.182
x ⫺ ␮ 53,220 ⫺ 52,200 1020
⫽
⫽
⫽ 1.7
s
1200
600
冪n
冪4
Fail to reject H0 . There is not enough evidence to support the claim.
t⫽
23. (a) H0: ␮ > 100 ; Ha : ␮ < 100 (claim)
(b) t0 ⫽ ⫺3.747; Reject H0 if t < ⫺3.747.
(c) t ⫽
x ⫺ ␮ 75 ⫺ 100
⫺25
⫽
⫽
⬇ ⫺4.472
s
12.50
5.590
冪n
冪5
(d) Reject H0.
(e) There is sufficient evidence at the 1% significance level to support the claim that the
mean repair cost for damaged microwave ovens is less than $100.
24. (a) H0 : ␮ ⱕ $95; Ha : ␮ > $95 (claim)
(b) t0 ⫽ 3.143; Reject H0 if t > 3.143.
(c) t ⫽
x ⫺ ␮ 100 ⫺ 95
5
⫽
⫽
⬇ 0.311
s
42.50
16.0635
冪n
冪7
(d) Fail to reject H0 .
(e) There is not enough evidence at the 1% significance level to support the claim that the
mean repair cost for damaged computers is more than $95.
25. (a) H0: ␮ ⱕ 1; Ha : ␮ > 1 (claim)
(b) t0 ⫽ 1.796; Reject H0 if t > 1.796.
(c) t ⫽
x ⫺ ␮ 1.46 ⫺ 1
0.46
⫽
⫽
⬇ 5.691
s
0.28
0.081
冪n
冪12
(d) Reject H0.
(e) There is sufficient evidence at the 5% significance level to support the claim that the
mean waste recycled by adults in the United States is more than 1 pound per person
per day.
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26. (a) H0 : ␮ ⱕ 4; Ha : ␮ > 4 (claim)
(b) t0 ⫽ 1.833; Reject H0 if t > 1.833.
(c) t ⫽
x ⫺ ␮ 4.54 ⫺ 4
0.54
⫽
⫽
⫽ 1.411
s
1.21
0.383
冪n
冪10
(d) Fail to reject H0 .
(e) There is not enough evidence at the 5% significance level to support the claim that the
mean waste generated by adults in the U.S. is more than 4 pounds per day.
27. (a) H0: ␮ ⫽ $25,000 (claim); Ha: ␮ ⫽ $25,000
(b) t0 ⫽ ± 2.262; Reject H0 if t < ⫺2.262 or t > 2.262.
(c) x ⬇ 25,852.2
t⫽
s ⬇ $3197.1
x ⫺ ␮ 25,852.2 ⫺ 25,000 ⫺852.2
⫽
⫽
⬇ 0.843
s
3197.1
1011.0
冪n
冪10
(d) Fail to reject H0.
(e) There is insufficient evidence at the 5% significance level to reject the claim that the
mean salary for full-time male workers over age 25 without a high school diploma is
$25,000.
28. (a) H0 : ␮ ⫽ $19,100 (claim); Ha : ␮ ⫽ $19,100
(b) t0 ⫽ ± 2.201; Reject H0 if t < ⫺2.201 or t > 2.201.
(c) x ⬇ $18,886.5,
t⫽
s ⬇ $1397.4
x ⫺ ␮ 18,886.5 ⫺ 19,100 ⫺213.5
⫽
⫽
⬇ ⫺0.529
s
1397.4
403.4
冪n
冪12
(d) Fail to reject H0 .
(e) There is not enough evidence at the 5% significance level to reject the claim that the
mean annual pay for full-time female workers over age 25 without high school diplomas
is $19,100.
29. (a) H0: ␮ ⱖ 3.0; Ha: ␮ < 3.0 (claim)
(b) x ⫽ 1.925
t⫽
x ⫽ 0.654
x ⫺ ␮ 1.925 ⫺ 3.0 ⫺1.075
⫽
⫽
⬇ ⫺7.351
s
0.654
0.146
冪n
冪20
P-value ⫽ 再Area left of t ⫽ ⫺7.351冎 ⬇ 0
(c) Reject H0.
(e) There is sufficient evidence at the 5% significance level to support the claim that
teenage males drink fewer than three 12-ounce servings of soda per day.
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HYPOTHESIS TESTING WITH ONE SAMPLE
30. (a) H0 : ␮ ⱕ $550; Ha : ␮ > $550 (claim)
(b) x ⫽ 605,
t⫽
s ⫽ 150.8
x ⫺ ␮ 605 ⫺ 550
55
⫽
⫽
⬇ 1.787
s
150.8
30.782
冪n
冪24
P-value ⫽ 再Area right of t ⫽ 1.787冎 ⫽ 0.0436
(c) Reject H0.
(d) There is sufficient evidence at the 5% significance level to support the claim that teachers spend a mean of more than $550 of their own money on school supplies in a year.
31. (a) H0 : ␮ ⱖ 32; Ha : ␮ < 32 (claim)
(b) x ⫽ 30.167
t⫽
s ⫽ 4.004
x ⫺ ␮ 30.167 ⫺ 32 ⫺1.833
⫽
⫽
⬇ ⫺1.942
s
4.004
0.944
冪n
冪18
P-value ⫽ 再Area left of t ⫽ ⫺1.942冎 ⬇ 0.0344
(c) Fail to reject H0.
(e) There is insufficient evidence at the 1% significance level to support the claim that the
mean class size for full-time faculty is fewer than 32.
32. (a) H0 : ␮ ⫽ 11.0 (claim); Ha : ␮ ⫽ 11.0
(b) x ⫽ 10.050, s ⫽ 2.485
t⫽
x ⫺ ␮ 10.050 ⫺ 11.0
⫺.95
⫽
⫽
⬇ ⫺1.081
s
2.485
0.879
冪n
冪8
P-value ⫽ 2再area left of t ⫽ ⫺1.081冎 ⫽ 2共0.15775兲 ⫽ 0.3155
(c) Fail to reject H0 .
(d) There is not enough evidence at the 1% significance level to reject the claim that the
mean number of classroom hours per week for full-time faculty is 11.0.
33. (a) H0 : ␮ ⫽ $2634 (claim); Ha : ␮ ⫽ $2634
(b) x ⫽ $2785.6
t⫽
s ⫽ $759.3
x ⫺ ␮ 2785.6 ⫺ 2634
151.6
⫽
⫽
⫽ 0.692
s
759.3
219.19
冪n
冪12
P-value ⫽ 2再Area right of t ⫽ 0.692冎 ⫽ 2再0.2518冎 ⫽ 0.5036
(c) Fail to reject H0.
(e) There is insufficient evidence at the 2% significance level to reject the claim that the typical
household in the U.S. spends a mean amount of $2634 per year on food away from home.
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34. (a) H0 : ␮ ⫽ $152 (claim); Ha : ␮ ⫽ $152
(b) x ⫽ $142.8, s ⫽ $37.52
t⫽
x ⫺ ␮ 142.8 ⫺ 152
⫺9.2
⫽
⫽
⫽ ⫺0.775
s
37.52
11.865
冪n
冪10
P-value ⫽ 2再Area left of t ⫽ ⫺0.78冎
⫽ 2再0.229冎
⫽ 0.4580
(c) Fail to reject H0.
(d) There is insufficient evidence at the 2% significance level to reject the claim that the
daily lodging costs for a family in the U.S. is $152.
35. H0 : ␮ ⱕ $2328; Ha : ␮ > 2328 (claim)
t⫽
x ⫺ ␮ 2528 ⫺ 2328
200
⫽
⫽
⬇ 1.507
s
325
132.681
冪n
冪6
P-value ⫽ 再Area right of t ⫽ 1.507冎 ⬇ 0.096
Because 0.096 > 0.01 ⫽ ␣, fail to reject H0 .
36. (a) Because 0.096 > 0.05 ⫽ ␣, fail to reject H0.
(b) Because 0.096 < 0.10 ⫽ ␣, reject H0.
(c) t ⫽
x ⫺ ␮ 2528 ⫺ 2328
200
⫽
⫽
⫽ 2.132
s
325
93.819
冪n
冪12
P-value ⫽ 再Area right of t ⫽ 2.132冎 ⬇ 0.028
Because 0.028 > 0.01 ⫽ ␣, fail to reject H0.
(d) t ⫽
200
x ⫺ ␮ 2528 ⫺ 2328
⫽
⫽
⫽ 3.015
s
325
66.340
冪n
冪24
P-value ⫽ 再Area right of t ⫽ 3.015冎 ⬇ 0.003
Since 0.003 < 0.01 ⫽ ␣, reject H0.
37. Because ␴ is unknown, n < 30, and the gas mileage is normally distributed, use
the t-distribution.
H0 : ␮ ⱖ 23 (claim); Ha : ␮ < 23
t⫽
x ⫺ ␮ 22 ⫺ 23
⫺1
⫽
⫽
⬇ ⫺0.559
s
4
1.789
冪n
冪5
P-value ⫽ 再Area left of t ⫽ ⫺0.559冎 ⫽ 0.303
Fail to reject H0. There is insufficient evidence at the 5% significance level to reject the
claim that the mean gas mileage for the luxury sedan is at least 23 miles per gallon.
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HYPOTHESIS TESTING WITH ONE SAMPLE
38. Because ␴ is unknown and n ⱖ 30, use the z-distribution.
H0 : ␮ ⱖ 23,000; Ha : ␮ < 23,000 (claim)
z⫽
x ⫺ ␮ 21,856 ⫺ 23,000 ⫺1144
⫽
⫽
⬇ ⫺2.557
s
3163
447.32
冪n
冪50
P-value ⫽ 2再Area left of z ⫽ ⫺2.557冎 ⫽ 2共0.0026兲 ⫽ 0.0052
Reject H0 . There is enough evidence at the 1% significance level to reject the claim that
the mean price for 1 year of graduate school for a full-time student in a master’s degree
program at a public institution is less than $23,000.
7.4 HYPOTHESIS TESTING FOR PROPORTIONS
7.4 Try It Yourself Solutions
1a. np ⫽ 共86兲共0.30兲 ⫽ 25.8 > 5, nq ⫽ 共86兲共0.70兲 ⫽ 60.2 > 5
b. The claim is “less than 30% of cellular phone users whose phone can connect to the
Internet have done so while at home.”
H0 : p ⱖ 0.30; Ha : p < 0.30 (claim)
c. ␣ ⫽ 0.05
d. z0 ⫽ ⫺1.645; Reject H0 if z < ⫺1.645.
p⫺p
^
e. z ⫽
⫽
0.20 ⫺ 0.30
冪pqn 冪共0.3086兲共0.70兲
⫽
⫺0.1
⬇ ⫺2.024
0.0494
f. Reject H0 .
g. There is enough evidence to support the claim.
2a. np ⫽ 共250兲共0.05兲 ⫽ 12.5 > 5, nq ⫽ 共250兲共0.95兲 ⫽ 237.5 > 5
b. The claim is “5% of U.S. adults have had vivid dreams about UFOs.”
H0 : p ⫽ 0.05 (claim); Ha: p ⫽ 0.05
c. ␣ ⫽ 0.01
d. z0 ⫽ ± 2.575; Reject H0 if z < ⫺2.575 or z > 2.575.
p⫺p
^
e. z ⫽
⫽
0.08 ⫺ 0.05
兲共0.95兲
冪pqn 冪共0.05250
⫽
0.03
⬇ 2.176
0.0138
f. Fail to reject H0 .
g. There is not enough evidence to reject the claim.
3a. np ⫽ 共75兲共0.30兲 ⫽ 22.5 > 5, nq ⫽ 共75兲共0.70兲 ⫽ 52.5 > 5
b. The claim is “more than 30% of U.S. adults regularly watch the Weather Channel.”
H0 : p ⱕ 0.30; Ha: p > 0.30 (claim)
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c. ␣ ⫽ 0.01
d. z0 ⫽ 2.33; Reject H0 if z > 2.33.
x
27
⫽
⫽ 0.360
n 75
p⫺p
0.360 ⫺ 0.30
0.06
z⫽
⫽
⫽
⬇ 1.13
0.053
pq
共0.30兲共0.70兲
n
75
e. p ⫽
^
^
冪
冪
f. Fail to reject H0 .
g. There is not enough evidence to support the claim.
7.4 EXERCISE SOLUTIONS
1. Verify that np ⱖ 5 and nq ⱖ 5. State H0 and Ha . Specify the level of significance ␣.
Determine the critical value(s) and rejection region(s). Find the standardized test statistic.
Make a decision and interpret in the context of the original claim.
2. If np ⱖ 5 and nq ⱖ 5, the normal distribution can be used.
3. np ⫽ 共105兲共0.25兲 ⫽ 26.25 > 5
nq ⫽ 共105兲共0.75兲 ⫽ 78.75 > 5 → use normal distribution
H0 : p ⫽ 0.25; Ha : p ⫽ 0.25 (claim)
z0 ± 1.96
p⫺p
^
z⫽
⫽
0.239 ⫺ 0.25
兲共0.75兲
冪pqn 冪共0.25105
⫽
⫺0.011
⬇ ⫺0.260
0.0423
Fail to reject H0 . There is not enough evidence to support the claim.
4. np ⫽ 共500兲共0.30兲 ⫽ 150 ⱖ 5
nq ⫽ 共500兲共0.70兲 ⫽ 350 ⱖ 5 → use normal distribution
H0 : p ⱕ 0.30 (claim); Ha : p > 0.30
z0 ⫽ 1.645
p⫺p
^
z⫽
⫽
0.35 ⫺ 0.30
兲共0.70兲
冪pqn 冪共0.30500
⫽
0.05
⬇ 2.440
0.0205
Reject H0 . There is enough evidence to reject the claim.
5. np ⫽ 共20兲共0.12兲 ⫽ 2.4 < 5
nq ⫽ 共20兲共0.88兲 ⫽ 17.6 ⱖ 5 → cannot use normal distribution
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HYPOTHESIS TESTING WITH ONE SAMPLE
6. np ⫽ 共45兲共0.125兲 ⫽ 5.625 ⱖ 5
nq ⫽ 共45兲共0.875兲 ⫽ 39.375 ⱖ 5 → use normal distribution
H0 : p ⱕ 0.125; Ha : p > 0.125 (claim)
z0 ⫽ 2.33
p⫺p
^
z⫽
⫽
0.2325 ⫺ 0.125
冪pqn 冪共0.12545兲共0.875兲
⫽
0.1075
⬇ 2.180
0.0493
Fail to reject H0. There is not enough evidence to support the claim.
7. np ⫽ 共70兲共0.48兲 ⫽ 33.6 ⱖ 5
nq ⫽ 共70兲共0.52兲 ⫽ 36.4 ⱖ 5 → use normal distribution
H0 : p ⱖ 0.48 (claim); Ha : p < 0.48
z0 ⫽ ⫺1.29
p⫺p
^
z⫽
⫽
0.40 ⫺ 0.48
冪pzn 冪共0.4870兲共0.52兲
⫽
⫺0.08
⬇ ⫺1.34
0.060
Reject H0 . There is enough evidence to reject the claim.
8. np ⫽ 共16兲共0.80兲 ⫽ 12.8 ⱖ 5
nq ⫽ 共16兲共0.20兲 ⫽ 3.2 < 5 → cannot use normal distribution
9. (a) H0 : p ⱖ 0.20 (claim); Ha : p < 0.20
(b) z0 ⫽ ⫺2.33; Reject H0 if z < ⫺2.33.
p⫺p
^
(c) z ⫽
⫽
0.185 ⫺ 0.20
兲共0.80兲
冪pqn 冪共0.20200
⫽
⫺0.015
⬇ ⫺0.53
0.0283
(d) Fail to reject H0.
(e) There is insufficient evidence at the 1% significance level to reject the claim that at least
20% of U.S. adults are smokers.
10. (a) H0 : p ⱕ 0.40 (claim); Ha : p > 0.40
(b) z0 ⫽ 2.33; Reject H0 if z > 2.33.
p⫺p
^
(c) z ⫽
⫽
0.416 ⫺ 0.40
兲共0.60兲
冪pqn 冪共0.40250
⫽
0.016
⬇ 0.52
0.0310
(d) Fail to reject H0 .
(e) There is not enough evidence at the 1% significance level to reject the claim that no
more than 40% of U.S. adults eat breakfast every day.
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11. (a) H0 : p ⱕ 0.30; Ha : p > 0.30 (claim)
(b) z0 ⫽ 1.88; Reject H0 if z > 1.88.
p⫺p
0.32 ⫺ 0.3
^
(c) z ⫽
⫽
兲共0.70兲
冪pqn 冪共0.301050
⫽
0.02
⬇ 1.41
0.0141
(d) Fail to reject H0.
(e) There is insufficient evidence at the 3% significance level to support the claim that
more than 30% of U.S. consumers have stopped buying the product because the manufacturing of the product pollutes the environment.
12. (a) H0 : p ⱕ 0.60; Ha : p > 0.60 (claim)
(b) z0 ⫽ 1.28; Reject H0 if z > 1.28.
p⫺p
^
(c) z ⫽
⫽
0.65 ⫺ 0.60
兲共0.40兲
冪pqn 冪共0.60100
⫽
0.05
⫽ 1.02
0.0490
(d) Fail to reject H0 .
(e) There is not enough evidence at the 10% significance level to support the claim
that more than 60% of British consumers are concerned about the use of genetic
modification in food production and want to avoid genetically modified foods.
13. (a) H0 : p ⫽ 0.44 (claim); Ha : p ⫽ 0.44
(b) z0 ⫽ ± 2.33; Reject H0 if z < ⫺2.33 or z > 2.33.
(c) p ⫽
^
722
⬇ 0.410
1762
p⫺p
^
z⫽
⫽
0.410 ⫺ 0.44
兲共0.56兲
冪pqn 冪共0.441762
⫽
⫺0.03024
⬇ ⫺2.537
0.01183
(d) Reject H0.
(e) There is sufficient evidence at the 2% significance level to reject the claim that 44% of
home buyers find their real estate agent through a friend.
14. (a) H0 : p ⫽ 0.24 (claim); Ha : p ⫽ 0.24
(b) z0 ⫽ ± 1.96; Reject H0 if z < ⫺1.96 or z > 1.96.
(c) p ⫽
^
292
⬇ 0.2716
1075
p⫺p
^
z⫽
⫽
0.2716 ⫺ 0.240
兲共0.760兲
冪pqn 冪共0.2401075
⫽
0.316
⬇ 2.43
0.013
(d) Reject H0 .
(e) There is sufficient evidence at the 5% significance level to reject the claim that 24% of
adults in the United States are afraid to fly.
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HYPOTHESIS TESTING WITH ONE SAMPLE
15. H0 : p ⱖ 0.52 (claim); Ha : p < 0.52
z0 ⫽ ⫺1.645; Rejection region: z < ⫺1.645
p⫺p
^
z⫽
⫽
0.48 ⫺ 0.52
冪pqn 冪共0.5250兲共0.48兲
⫽
⫺0.04
⬇ ⫺0.566
0.0707
Fail to reject H0 . There is insufficient evidence to reject the claim.
16. The company should continue the use of giveaways because there is not enough evidence to
say that less than 52% of the adults would be more likely to buy a product when there are
free samples.
17. H0 : p ⫽ 0.44 (claim); Ha : p ⫽ 0.44
z⫽
x ⫺ np
722 ⫺ 共1762兲共0.44兲
⫺53.28
⫽
⫽
⬇ ⫺2.56
20.836
冪npq
冪共1762兲共0.44兲共0.56兲
Reject H0. The results are the same.
p⫺p
^
18. z ⫽
冪pqn
⇒
冢nx 冣 ⫺ p
冪pqn
⇒
冢nx 冣 ⫺ p
冪pq
冤
冢nx 冣 ⫺ p冥
⇒
冪pq
冪n
x
⫺ p冥 n
冤
冢
n
n冣
⇒
冪
⭈冪
n
冪pqn
⇒
x ⫺ np
冪pqn
冪n
7.5 HYPOTHESIS TESTING FOR VARIANCE
AND STANDARD DEVIATION
7.5 Try It Yourself Solutions
1a. ␹ 20 ⫽ 33.409
2a. ␹ 20 ⫽ 17.708
3a. ␹ R2 ⫽ 31.526
b. ␹ L2 ⫽ 8.231
4a. The claim is “the variance of the amount of sports drink in a 12-ounce bottle is no more
than 0.40.”
H0 : ␴ 2 ⱕ 0.40 (claim); Ha : ␴ 2 > 0.40
b. ␣ ⫽ 0.01 and d.f. ⫽ n ⫺ 1 ⫽ 30
c. ␹ 20 ⫽ 50.892; Reject H0 if ␹ 2 > 50.892.
共n ⫺ 1兲s2 共30兲共0.75兲
⫽
⫽ 56.250
␴2
0.40
e. Reject H0 .
d. ␹ 2 ⫽
f. There is enough evidence to reject the claim.
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5a. The claim is “the standard deviation in the length of response times is less than
3.7 minutes.”
H0 : ␴ ⱖ 3.7; Ha : ␴ < 3.7 (claim)
b. ␣ ⫽ 0.05 and d.f. ⫽ n ⫺ 1 ⫽ 8
c. ␹ 20 ⫽ 2.733; Reject H0 if ␹ 2 < 2.733.
d. ␹ 2 ⫽
共n ⫺ 1兲s2 共8兲共3.0兲2
⫽
⬇ 5.259
␴2
共3.7兲2
e. Fail to reject H0 .
f. There is not enough evidence to support the claim.
6a. The claim is “the variance of the diameters in a certain tire model is 8.6.”
H0 : ␴ 2 ⫽ 8.6 (claim); Ha : ␴ 2 ⫽ 8.6
b. ␣ ⫽ 0.01 and d.f. ⫽ n ⫺ 1 ⫽ 9
c. ␹ 2L ⫽ 1.735 and ␹ R2 ⫽ 23.589
Reject H0 if ␹ 2 > 23.589 or ␹ 2 < 1.735.
d. ␹ 2 ⫽
共n ⫺ 1兲s 2 共9兲共4.3兲
⫽
⫽ 4.50
␴2
共8.6兲
e. Fail to reject H0 .
f. There is not enough evidence to reject the claim.
7.5 EXERCISE SOLUTIONS
1. Specify the level of significance ␣. Determine the degrees of freedom. Determine the
critical values using the ␹ 2 distribution. If (a) right-tailed test, use the value that corresponds
to d.f. and ␣ . (b) left-tailed test, use the value that corresponds to d.f. and 1 ⫺ ␣ ; and
1
1
(c) two-tailed test, use the value that corresponds to d.f. and 2␣ and 1 ⫺ 2␣.
2. State H0 and Ha . Specify the level of significance. Determine the degrees of freedom.
Determine the critical value(s) and rejection region(s). Find the standardized test statistic.
Make a decision and interpret in the context of the original claim.
3. ␹ 20 ⫽ 38.885
4. ␹ 20 ⫽ 14.684
7. ␹ L2 ⫽ 7.261, ␹ R2 ⫽ 24.996
5. ␹ 20 ⫽ 0.872
6. ␹ 20 ⫽ 13.091
8. ␹L2 ⫽ 12.461, ␹R2 ⫽ 50.993
9. (a) Fail to reject H0.
10. (a) Fail to reject H0.
(b) Fail to reject H0.
(b) Fail to reject H0.
(c) Fail to reject H0.
(c) Reject H0.
(d) Reject H0.
(d) Reject H0.
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11. (a) Fail to reject H0.
|
HYPOTHESIS TESTING WITH ONE SAMPLE
12. (a) Fail to reject H0.
(b) Reject H0.
(b) Fail to reject H0.
(c) Reject H0.
(c) Fail to reject H0.
(d) Fail to reject H0.
(d) Reject H0.
13. H0: ␴ 2 ⫽ 0.52 (claim); Ha: ␴ 2 ⫽ 0.52
␹ L2 ⫽ 7.564, ␹ R2 ⫽ 30.191
␹2 ⫽
共n ⫺ 1兲s2 共17兲共0.508兲2
⫽
⬇ 16.608
␴2
共0.52兲
Fail to reject H0. There is insufficient evidence to reject the claim.
14. H0: ␴ ⱖ 40; Ha: ␴ < 40 (claim)
␹ 20 ⫽ 3.053
␹2 ⫽
共n ⫺ 1兲s2 共11兲共40.8兲2
⫽
⬇ 11.444
␴2
共40兲2
Fail to reject H0 . There is insufficient evidence to support the claim.
15. (a) H0: ␴ 2 ⫽ 3 (claim); Ha: ␴ 2 ⫽ 3
(b) ␹ L2 ⫽ 13.844, ␹ R2 ⫽ 41.923; Reject H0 if ␹ 2 > 41.923 or ␹ 2 < 13.844.
(c) ␹ 2 ⫽
共n ⫺ 1兲s2 共26兲共2.8兲
⫽
⬇ 24.267
␴2
3
(d) Fail to reject H0.
(e) There is insufficient evidence at the 5% level of significance to reject the claim that the
variance of the life of the appliances is 3.
16. (a) H0 : ␴ 2 ⫽ 6 (claim); Ha : ␴ 2 ⫽ 6
(b) ␹L2 ⫽ 14.573, ␹R2 ⫽ 43.194; Reject H0 if ␹ 2 > 43.194 or ␹ 2 < 14.573.
(c) ␹ 2 ⫽
共n ⫺ 1兲s2 共27兲共4.25兲
⫽
⫽ 19.125
␴2
6
(d) Fail to reject H0 .
(e) There is not enough evidence at the 5% significance level to reject the claim that the
variance of the gas mileage is 6.
17. (a) H0: ␴ ⱖ 36; Ha: ␴ < 36 (claim)
(b) ␹ 20 ⫽ 13.240; Reject H0 if ␹ 2 < 13.240.
(c) ␹ 2 ⫽
共n ⫺ 1兲s2 共21兲共33.4兲2
⫽
⬇ 18.076
␴2
共36兲2
(d) Fail to reject H0.
(e) There is insufficient evidence at the 10% significance level to support the claim that the
standard deviation for eighth graders on the examination is less than 36.
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18. (a) H0 : ␴ ⱖ 30; Ha : ␴ < 30 (claim)
(b) ␹ 20 ⫽ 6.408; Reject H0 if ␹ 2 < 6.408.
(c) ␹ 2 ⫽
共n ⫺ 1兲s2 共17兲共33.6兲2
⫽
⬇ 21.325
␴2
共30兲2
(d) Fail to reject H0 .
(e) There is not enough evidence at the 1% significance level to support the claim that the
standard deviation of test scores for eighth grade students who took a U.S. history
assessment test is less than 30 points.
19. (a) H0: ␴ ⱕ 0.5 (claim); Ha: ␴ > 0.5
(b) ␹ 20 ⫽ 33.196; Reject H0 if ␹ 2 > 33.196.
(c) ␹ 2 ⫽
共n ⫺ 1兲s2 共24兲共0.7兲2
⫽
⫽ 47.04
␴2
共0.5兲2
(d) Reject H0.
(e) There is sufficient evidence at the 10% significance level to reject the claim that the
standard deviation of waiting times is no more than 0.5 minute.
20. (a) H0 : ␴ ⫽ 6.14 (claim); Ha: ␴ ⫽ 6.14
(b) ␹ L2 ⫽ 8.907, ␹R2 ⫽ 32.852; Reject H0 if ␹2 < 8.907 or ␹ 2 > 32.852.
(c) ␹ 2 ⫽
共n ⫺ 1兲s 2 共19兲共6.5兲2
⫽
⫽ 21.293
␴2
共6.14兲2
(d) Fail to reject H0 .
(e) There is insufficient evidence at the 5% significance level to reject the claim that the
standard deviation of the lengths of stay is 6.14 days.
21. (a) H0: ␴ ⱖ $3500; Ha: ␴ < $3500 (claim)
(b) ␹ 20 ⫽ 18.114; Reject H0 if ␹ 2 < 18.114.
(c) ␹ 2 ⫽
共n ⫺ 1兲s2 共27兲共4100兲2
⫽
⬇ 37.051
␴2
共3500兲2
(d) Fail to reject H0.
(e) There is insufficient evidence at the 10% significance level to support the claim that the
standard deviation of the total charge for patients involved in a crash where the vehicle
struck a construction baracade is less than $3500.
22. (a) H0 : ␴ ⱕ $30 (claim); Ha : ␴ > $30
(b) ␹ 20 ⫽ 37.566; Reject H0 if ␹ 2 < 37.566.
(c) ␹ 2 ⫽
共n ⫺ 1兲s2 共20兲共35.25兲2
⫽
⫽ 27.613
␴2
共30兲2
(d) Fail to reject H0 .
(e) There is not enough evidence at the 1% significance level to reject the claim that the
standard deviation of the room rates of hotels in the city is no more than $30.
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HYPOTHESIS TESTING WITH ONE SAMPLE
23. (a) H0: ␴ ⱕ $20,000; Ha: ␴ > $20,000 (claim)
(b) ␹ 20 ⫽ 24.996; Reject H0 if ␹ 2 > 24.996.
(c) s ⫽ 20,826.145
␹2 ⫽
共n ⫺ 1兲s 2 共15兲共20,826.145兲2
⫽
⬇ 16.265
␴2
共20,000兲2
(d) Fail to reject H0 .
(e) There is insufficient evidence at the 5% significance level to support the claim that the
standard deviation of the annual salaries for actuaries is more than $20,000.
24. (a) H0 : ␴ ⱖ $14,500 (claim); Ha : ␴ < $14,500
(b) ␹ 20 ⫽ 10.085; Reject H0 if ␹ 2 < 10.085.
(c) s ⫽ 13,950.604
␹2 ⫽
共n ⫺ 1兲s2 共17兲共13,950.604兲2
⫽
⬇ 15.736
␴2
共14,500兲2
(d) Fail to reject H0 .
(e) There is not enough evidence at the 10% significance level to reject the claim that
the standard deviation of the annual salaries for public relations managers is at least
$14,500.
25. ␹ 2 ⫽ 37.051
P-value ⫽ 再Area left of ␹ 2 ⫽ 37.051冎 ⫽ 0.9059
Fail to reject H0 because P-value ⫽ 0.9059 > 0.10 ⫽ ␣.
26. ␹ 2 ⫽ 27.613
P-value ⫽ 再Area right of ␹ 2 ⫽ 27.613冎 ⫽ 0.1189
Fail to reject H0 because P-value ⫽ 0.1189 > 0.01 ⫽ ␣.
27. ␹ 2 ⫽ 16.265
P-value ⫽ 再Area right of ␹ 2 ⫽ 16.265冎 ⫽ 0.3647
Fail to reject H0 because P-value ⫽ 0.3647 > 0.05 ⫽ ␣.
28. ␹ 2 ⫽ 15.736
P-value ⫽ 再Area left of ␹ 2 ⫽ 15.736冎 ⫽ 0.4574
Fail to reject H0 because P-value ⫽ 0.4574 > 0.10 ⫽ ␣.
CHAPTER 7 REVIEW EXERCISE SOLUTIONS
1. H0 : ␮ ⱕ 1479 (claim); Ha : ␮ > 1479
2. H0 : ␮ ⫽ 95 (claim); Ha : ␮ ⫽ 95
3. H0 : p ⱖ 0.205; Ha : p < 0.205 (claim)
4. H0 : ␮ ⫽ 150,020; Ha : ␮ ⫽ 150,020 (claim)
5. H0 : ␴ ⱕ 6.2; Ha : ␴ > 6.2 (claim)
6. H0 : p ⱖ 0.78 (claim); Ha : p < 0.78
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7. (a) H0 : p ⫽ 0.73 (claim); Ha : p ⫽ 0.73
(b) Type I error will occur if H0 is rejected when the actual proportion of college students
that occasionally or frequently come late to class is 0.63.
Type II error if H0 is not rejected when the actual proportion of college students that
occasionally or frequently come late to class is not 0.63.
(c) Two-tailed, because hypothesis compares “⫽ vs ⫽”.
(d) There is enough evidence to reject the claim.
(e) There is not enough evidence to reject the claim.
8. (a) H0 : ␮ ⱖ 30,000 (claim); Ha : ␮ < 30,000
(b) Type I error will occur if H0 is rejected when the actual mean tire life is at least
30,000 miles.
Type II error if H0 is not rejected when the actual mean tire life is less than
30,000 miles.
(c) Left-tailed, because hypothesis compares “ ⱖ vs <”.
(d) There is enough evidence to reject the claim.
(e) There is not enough evidence to reject the claim.
9. (a) H0 : ␮ ⱕ 50 (claim); Ha : ␮ > 50
(b) Type I error will occur if H0 is rejected when the actual standard deviation sodium
content is no more than 50 milligrams.
Type II error if H0 is not rejected when the actual standard deviation sodium content is
more than 50 milligrams.
(c) Right-tailed, because hypothesis compares “ ⱕ vs >”.
(d) There is enough evidence to reject the claim.
(e) There is not enough evidence to reject the claim.
10. (a) H0 : ␮ ⱖ 25; Ha : ␮ < 25 (claim)
(b) Type I error will occur if H0 is rejected when the actual mean number of grams of
carbohydrates in one bar is greater than or equal to 25.
Type II error if H0 is not rejected when the actual mean number of grams of
carbohydrates in one bar is less than 25.
(c) Left-tailed, because hypothesis compares “ ⱖ vs <”.
(d) There is enough evidence to support the claim.
(e) There is not enough evidence to support the claim.
11. z0 ⬇ ⫺2.05
12. z0 ⫽ ± 2.81
13. z0 ⫽ 1.96
14. z0 ⫽ ± 1.75
15. H0 : ␮ ⱕ 45 (claim); Ha : ␮ > 45
z0 ⫽ 1.645
z⫽
x ⫺ ␮ 47.2 ⫺ 45
2.2
⫽
⫽
⬇ 2.128
s
6.7
1.0338
冪n
冪42
Reject H0 . There is enough evidence to reject the claim.
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CHAPTER 7
|
HYPOTHESIS TESTING WITH ONE SAMPLE
16. H0 : ␮ ⫽ 0; Ha : ␮ ⫽ 0 (claim)
z0 ⫽ ± 1.96
z⫽
x ⫺ ␮ ⫺0.69 ⫺ 0 ⫺0.69
⫽
⫽
⬇ ⫺2.040
s
2.62
0.338
冪n
冪60
Reject H0. There is enough evidence to support the claim.
17. H0: ␮ ⱖ 5.500; Ha: ␮ < 5.500 (claim)
z0 ⫽ ⫺2.33
z⫽
x ⫺ ␮ 5.497 ⫺ 5.500
⫺0.003
⫽
⫽
⬇ ⫺1.636
s
0.011
0.00183
冪n
冪36
Fail to reject H0. There is not enough evidence to support the claim.
18. H0 : ␮ ⫽ 7450 (claim); Ha : ␮ ⫽ 7450
z0 ⫽ ± 1.96
z⫽
x ⫺ ␮ 7512 ⫺ 7450
62
⫽
⫽
⬇ 1.926
s
243
32.186
冪n
冪57
Fail to reject H0 . There is not enough evidence to reject the claim.
19. H0: ␮ ⱕ 0.05 (claim); Ha: ␮ > 0.05
z⫽
x ⫺ ␮ 0.057 ⫺ 0.05
0.007
⫽
⫽
⬇ 2.20
s
0.018
0.00318
冪n
冪32
P-value ⫽ 再Area right of z ⫽ 2.20冎 ⫽ 0.0139
␣ ⫽ 0.10; Reject H0.
␣ ⫽ 0.05; Reject H0.
␣ ⫽ 0.01; Fail to reject H0.
20. H0 : ␮ ⫽ 230; Ha : ␮ ⫽ 230 (claim)
z⫽
x ⫺ ␮ 216.5 ⫺ 230 ⫺13.5
⫽
⫽
⬇ ⫺5.41
s
17.3
2.497
冪n
冪48
P-value ⫽ 2再Area left of z ⫽ ⫺5.41冎 ⬇ 0
␣ ⫽ 0.10; Reject H0.
␣ ⫽ 0.05; Reject H0.
␣ ⫽ 0.01; Reject H0.
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21. H0: ␮ ⫽ 326 (claim); Ha: ␮ ⫽ 326
z⫽
x ⫺ ␮ 318 ⫺ 326
⫺8
⫽
⫽
⬇ ⫺2.263
s
25
3.536
冪n
冪50
P-value ⫽ 2再Area left of z ⫽ ⫺2.263冎 ⫽ 2再0.012冎 ⫽ 0.024
Reject H0. There is sufficient evidence to reject the claim.
22. H0 : ␮ ⱕ $650 (claim); Ha : ␮ > $650
z⫽
x ⫺ ␮ 657 ⫺ 650
7
⫽
⫽
⬇ 1.174
s
40
5.963
冪n
冪45
P-value ⫽ 再Area right of z ⫽ 1.17冎 ⫽ 0.1210
Fail to reject H0 .There is not enough evidence to reject the claim.
23. t0 ⫽ ± 2.093
24. t0 ⫽ 2.998
25. t0 ⫽ ⫺1.345
26. t0 ⫽ ± 2.201
27. H0: ␮ ⫽ 95; Ha: ␮ ⫽ 95 (claim)
t0 ⫽ ± 2.201
t⫽
x ⫺ ␮ 94.1 ⫺ 95
⫺0.9
⫽
⫽
⬇ ⫺2.038
s
1.53
0.442
冪n
冪12
Fail to reject H0. There is not enough evidence to support the claim.
28. H0 : ␮ ⱕ 12,700; Ha : ␮ > 12,700 (claim)
t0 ⫽ 1.725
t⫽
x ⫺ ␮ 12,804 ⫺ 12,700
104
⫽
⫽
⬇ 1.922
s
248
54.118
冪n
冪21
Reject H0. There is enough evidence to support the claim.
29. H0: ␮ ⱖ 0 (claim); Ha: ␮ < 0
t0 ⫽ ⫺1.341
t⫽
x ⫺ ␮ ⫺0.45 ⫺ 0 ⫺0.45
⬇ ⫺1.304
⫽
⫽
s
1.38
0.345
冪n
冪16
Fail to reject H0. There is not enough evidence to reject the claim.
30. H0 : ␮ ⫽ 4.20 (claim); Ha : ␮ ⫽ 4.20
t0 ⫽ ± 2.896
t⫽
x ⫺ ␮ 4.41 ⫺ 4.20
0.21
⫽
⫽
⬇ 2.423
s
0.26
0.0867
冪n
冪9
Fail to reject H0 . There is not enough evidence to reject the claim.
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CHAPTER 7
|
HYPOTHESIS TESTING WITH ONE SAMPLE
31. H0: ␮ ⱕ 48 (claim); Ha: ␮ > 48
t0 ⫽ 3.148
t⫽
x ⫺ ␮ 52 ⫺ 48
4
⫽
⫽
⬇ 4.233
s
2.5
0.945
冪n
冪7
Reject H0. There is enough evidence to reject the claim.
32. H0 : ␮ ⱖ 850; Ha : ␮ < 850 (claim)
t0 ⫽ ⫺2.160
t⫽
x ⫺ ␮ 875 ⫺ 850
25
⫽
⫽
⬇ 3.742
s
25
6.682
冪n
冪14
Fail to reject H0 . There is not enough evidence to support the claim.
33. H0 : ␮ ⫽ $25 (claim); Ha : ␮ ⫽ $25
t0 ⫽ ± 1.740
t⫽
x ⫺ ␮ 26.25 ⫺ 25
1.25
⫽
⫽
⬇ 1.642
s
3.23
0.761
冪n
冪18
Fail to reject H0 . There is not enough evidence to reject the claim.
34. H0 : ␮ ⱕ 10 (claim); Ha : ␮ > 10
t0 ⫽ 1.397
t⫽
x ⫺ ␮ 13.5 ⫺ 10
3.5
⫽
⫽
⬇ 1.810
s
5.8
1.933
冪n
冪9
Reject H0. There is enough evidence to reject the claim.
35. H0 : ␮ ⱖ $10,200 (claim); Ha: ␮ < $10,200
t0 ⫽ ⫺2.602
x ⫽ 9895.8
t⫽
s ⫽ 490.88
x ⫺ ␮ 9895.8 ⫺ 10,200 ⫺304.2
⫽
⫽
⬇ ⫺2.479
s
490.88
122.72
冪n
冪16
P-value ⬇ 0.0128
Fail to reject H0 . There is not enough evidence to reject the claim.
36. H0 : ␮ ⱕ 9; Ha : ␮ > 9 (claim)
x ⫽ 9.982,
t⫽
s ⫽ 2.125
x ⫺ ␮ 9.982 ⫺ 9 0.982
⫽
⫽
⬇ 1.532
s
2.125
0.641
冪n
冪11
P-value ⫽ 0.078
Fail to reject H0 . There is not enough evidence to support the claim.
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37. H0 : p ⫽ 0.15 (claim); Ha : p ⫽ 0.15
z0 ⫽ ± 1.96
p⫺p
^
z⫽
⫽
0.09 ⫺ 0.15
冪pqn 冪共0.1540兲共0.85兲
⫽
⫺0.06
⬇ ⫺1.063
0.0565
Fail to reject H0 . There is not enough evidence to reject the claim.
38. H0 : p ⱖ 0.70; Ha : p < 0.70 (claim)
z0 ⫽ ⫺2.33
p⫺p
^
z⫽
⫽
0.50 ⫺ 0.70
冪pqn 冪共0.7068兲共0.30兲
⫽
⫺0.2
⬇ ⫺3.599
0.0556
Reject H0 . There is enough evidence to support the claim.
39. Because np ⫽ 3.6 is less than 5, the normal distribution cannot be used to approximate the
binomial distribution.
40. H0 : p ⫽ 0.50 (claim); Ha : p ⫽ 0.50
z0 ⫽ ± 1.645
p⫺p
^
z⫽
⫽
0.71 ⫺ 0.50
兲共0.50兲
冪pqn 冪共0.50129
⫽
0.21
⬇ 4.770
0.0440
Reject H0 . There is enough evidence to reject the claim.
41. Because np ⫽ 1.2 < 5, the normal distribution cannot be used to approximate the
binomial distribution.
42. H0 : p ⫽ 0.34; Ha: p ⫽ 0.34 (claim)
z0 ⫽ ± 2.575
p⫺p
^
z⫽
⫽
0.29 ⫺ 0.34
冪pqn 冪共0.3460兲共0.66兲
⫽
⫺0.05
⬇ 0.820
0.061
Fail to reject H0 . There is not enough evidence to support the claim.
43. H0 : p ⫽ 0.20; Ha : p ⫽ 0.20 (claim)
z0 ⫽ ± 2.575
p⫺p
^
z⫽
⫽
0.23 ⫺ 0.20
冪pqn 冪共0.2056兲共0.80兲
⫽
0.03
⬇ 0.561
0.0534
Fail to reject H0 . There is not enough evidence to support the claim.
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CHAPTER 7
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HYPOTHESIS TESTING WITH ONE SAMPLE
44. H0 : p ⱕ 0.80 (claim); Ha : p > 0.80
z0 ⫽ 1.28
p⫺p
^
z⫽
⫽
0.85 ⫺ 0.80
冪pqn 冪共0.8043兲共0.20兲
⫽
0.05
⬇ 0.820
0.061
Fail to reject H0 . There is not enough evidence to reject the claim.
45. H0 : p ⱕ 0.40; Ha : p > 0.40 (claim)
z0 ⫽ 1.28
p⫽
^
x
1130
⫽
⬇ 0.414
n 2730
p⫺p
^
z⫽
⫽
0.414 ⫺ 0.40
兲共0.60兲
冪pqn 冪共0.402730
⫽
0.14
⬇ 1.493
0.0094
Reject H0 . There is enough evidence to support the claim.
46. H0 : p ⫽ 0.02 (claim); Ha : p ⫽ 0.02
z0 ⫽ ± 1.96
p⫽
^
x
3
⫽
⬇ 0.01
n 300
p⫺p
^
z⫽
⫽
0.01 ⫺ 0.02
兲共0.98兲
冪pqn 冪共0.02300
⫽
⫺0.01
⬇ ⫺1.24
0.0081
Fail to reject H0 . There is not enough evidence to reject the claim.
47. ␹ R2 ⫽ 30.144
48. ␹ L2 ⫽ 3.565, ␹ R2 ⫽ 29.819
49. ␹ R2 ⫽ 33.196
50. ␹ 20 ⫽ 1.145
51. H0 : ␴ 2 ≤ 2; Ha : ␴ 2 > 2 (claim)
␹ 20 ⫽ 24.769
␹2 ⫽
共n ⫺ 1兲s2 共17兲共2.95兲
⫽
⫽ 25.075
␴2
共2兲
Reject H0 . There is enough evidence to support the claim.
52. H0 : ␴ 2 ⱕ 60 (claim); Ha : ␴ 2 > 60
␹ 20 ⫽ 26.119
␹2 ⫽
共n ⫺ 1兲s2 共14兲共72.7兲
⫽
⬇ 16.963
␴2
共60兲
Fail to reject H0 . There is not enough evidence to reject the claim.
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211
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53. H0 : ␴ 2 ⫽ 1.25 (claim); Ha : ␴ 2 ⫽ 1.25
␹ L2 ⫽ 0.831, ␹ R2 ⫽ 12.833
␹2 ⫽
共n ⫺ 1兲s2 共5兲共1.03兲2
⫽
⬇ 3.395
␴2
共1.25兲2
Fail to reject H0. There is not enough evidence to reject the claim.
54. H0 : ␴ ⫽ 0.035; Ha : ␴ ⫽ 0.035 (claim)
␹L2 ⫽ 4.601, ␹R2 ⫽ 32.801
␹2 ⫽
共n ⫺ 1兲s2 共15兲共0.026兲2
⫽
⬇ 8.278
␴2
共0.035兲2
Fail to reject H0 . There is not enough evidence to support the claim.
55. H0 : ␴ 2 ⱕ 0.01 (claim); Ha : ␴ 2 > 0.01
␹ 20 ⫽ 49.645
␹2 ⫽
共n ⫺ 1兲s2 共27兲共0.064兲
⫽
⫽ 172.800
␴2
共0.01兲
Reject H0. There is enough evidence to reject the claim.
56. H0 : ␴ ⱕ 0.0025 (claim); Ha : ␴ > 0.0025
␹ 20 ⫽ 27.688
␹2 ⫽
共n ⫺ 1兲s2 共13兲共0.0031兲2
⫽
⬇ 19.989
␴2
共0.0025兲2
Fail to reject H0 . There is not enough evidence to reject the claim.
CHAPTER 7 QUIZ SOLUTIONS
1. (a) H0: ␮ ⱖ 22 (claim); Ha: ␮ < 22
(b) “ ⱖ vs <” → Left-tailed
␴ is unknown and n ⱖ 30 → z-test.
(c) z0 ⫽ ⫺2.05; Reject H0 if z < ⫺2.05.
(d) z ⫽
x ⫺ ␮ 21.6 ⫺ 22
⫺0.4
⬇ ⫺0.580
⫽
⫽
s
7
0.690
冪n
冪103
(e) Fail to reject H0. There is insufficient evidence at the 2% significance level to reject
the claim that the mean utilization of fresh citrus fruits by people in the U.S. is at least
22 pounds per year.
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CHAPTER 7
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HYPOTHESIS TESTING WITH ONE SAMPLE
2. (a) H0: ␮ ⱖ 20 (claim); Ha: ␮ < 20
(b) “ ⱖ vs <” → Left-tailed
␴ is unknown, the population is normal, and n < 30 → t-test.
(c) t0 ⫽ ⫺1.895; Reject H0 if t < ⫺1.895.
(d) z ⫽
x ⫺ ␮ 18 ⫺ 20
⫺2
⫽
⫽
⬇ ⫺1.131
s
5
1.768
冪n
冪8
(e) Fail to reject H0. There is insufficient evidence at the 5% significance level to reject the
claim that the mean gas mileage is at least 20 miles per gallon.
3. (a) H0 : p ⱕ 0.10 (claim); Ha : p > 0.10
(b) “ ⱕ vs >” → Right-tailed
np ⱖ 5 and nq ⱖ 5 → z-test
(c) z0 ⫽ 1.75; Reject H0 if z > 1.75.
p⫺p
^
(d) z ⫽
⫽
0.13 ⫺ 0.10
冪pqn 冪共0.1057兲共0.90兲
⫽
0.03
⬇ 0.75
0.0397
(e) Fail to reject H0. There is insufficient evidence at the 4% significance level to reject the
claim that no more than 10% of microwaves need repair during the first five years of use.
4. (a) H0 : ␴ ⫽ 113 (claim); Ha : ␴ ⫽ 113
(b) “⫽ vs ⫽ ’’ → Two-tailed
Assuming the scores are normally distributed and you are testing the hypothesized
standard deviation → ␹ 2 test.
(c) ␹ 2L ⫽ 3.565, ␹R2 ⫽ 29.819; Reject H0 if ␹ 2 < 3.565 or if ␹ 2 > 29.819.
(d) ␹ 2 ⫽
共n ⫺ 1兲s2 共13兲共108兲2
⫽
⬇ 11.875
␴2
共113兲2
(e) Fail to reject H0 . There is insufficient evidence at the 1% significance level to reject the
claim that the standard deviation of the SAT critical reading scores for the state is 105.
5. (a) H0 : ␮ ⫽ $48,718 (claim); Ha : ␮ ⫽ $48,718
(b) “⫽ vs ⫽” → Two-tailed
␴ is unknown, n < 30, and assuming the salaries are normally distributed → t-test.
(c) not applicable
(d) t ⫽
x ⫺ ␮ 47,164 ⫺ 48,718
⫺1554
⫽
⫽
⬇ ⫺0.828
s
6500
1876.388
冪n
冪12
P-value ⫽ 2再Area left of t ⫽ ⫺0.828冎 ⫽ 2共0.2126兲 ⫽ 0.4252
(e) Fail to reject H0. There is insufficient evidence at the 5% significance level to reject
the claim that the mean annual salary for full-time male workers ages 25 to 34 with a
bachelor’s degree is $48,718.
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213
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HYPOTHESIS TESTING WITH ONE SAMPLE
6. (a) H0 : ␮ ⫽ $201 (claim); Ha : ␮ ⫽ $201
(b) “⫽ vs ⫽” → Two-tailed
␴ is unknown, n ⱖ 30 → z-test.
(c) not applicable
(d) z ⫽
x ⫺ ␮ 216 ⫺ 201
15
⫽
⫽
⬇ 2.958
s
30
5.071
冪n
冪35
P-value ⫽ 2再Area right of z ⫽ 2.958冎 ⫽ 2再0.0015冎 ⫽ 0.0030
(e) Reject H0 . There is sufficient evidence at the 5% significance level to reject the claim
that the mean daily cost of meals and lodging for a family of four traveling in Kansas
is $201.
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No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.