290 CHAPTER 7 │ HYPOTHESIS TESTING WITH ONE SAMPLE
31.
()
2
24.5 20.25σ==
()
2
25.8 33.64s==
Hypothesis test results:
32.
()
2
2418 174,724σ==
()
2
2305 93,025s==
Hypothesis test results:
2
: variance of variable
σ
33. 237.051χ=
P-value = {Area left of 237.051χ=} = 0.9059
Fail to reject 0
H because P-value = 0.9059 > 0.10.
34. 227.613χ=
P-value = {Area right of 227.613χ=} = 0.1189
Fail to reject 0
H because P-value = 0.1189 > 0.01.
CHAPTER 7 │ HYPOTHESIS TESTING WITH ONE SAMPLE 291
CHAPTER 7 REVIEW EXERCISE SOLUTIONS
1. 0: 375 (claim); : 375
a
HHµµ≤> 2. 0: 82 (claim); : 82
a
HHµµ=≠
7a. 0: p 0.71 (claim); : 0.71
a
HHp=≠
b. A type I error will occur if the actual proportion of Americans who support plans to order deep
cuts in executive compensation at companies that have received federal bailout funds is 71%, but
you reject 0: 0.71Hp=.
A type II error will occur if the actual proportion is not 71%, but you fail to reject 0: 0.71Hp=.
8a. 0: 400 (claim); : 400
a
HHµµ≥<
b. A type I error will occur if the actual mean shelf life of the dried fruit is at least 400 days, but you
reject 0: 400Hµ≥.
A type II error will occur if the actual mean shelf life of the dried fruit is less than 400 days, but
you fail to reject 0: 400Hµ≥.
of the dried fruit is at least 400 days.
9a. 0: 50 (claim); : 50
a
HHσσ≤>
b. A type I error will occur if the actual standard deviation of the sodium content in one serving of a
certain soup is no more than 50 milligrams, but you reject 0: 50Hσ≤.
292 CHAPTER 7 │ HYPOTHESIS TESTING WITH ONE SAMPLE
10a. 0: 25; : 25 (claim)
a
HHµµ≥<
b. A type I error will occur if the actual mean number of grams of carbohydrates in one bar is at
least 25, but you reject 0: 25Hµ≥.
11. P-value = {Area to left of z = 0.94−} = 0.1736
Fail to reject 0
H.
12. P-value = 2{Area to right of z = 2.57} = 2(0.0051) = 0.0102
Reject 0
H.
14. 0: 230; : 230 (claim)
a
HHµµ=≠
216.5 230 5.41
17.3
48
x
zs
n
µ−−
== ≈−
P-value = 2{Area to left of z = 5.41−} 2(0)≈ = 0
0.10α=⇒ Reject 0
H.
0.05α=⇒ Reject 0
H.
0.01α=⇒ Reject 0
H.
CHAPTER 7 │ HYPOTHESIS TESTING WITH ONE SAMPLE 293
18. 00
1.75, 1.75zz−=− =
19. Fail to reject 0
H because 1.645 1.645z−<<.
20. Reject 0
H because 1.645z>.
23. 0: 45 (claim); : 45
a
HHµµ≤>
01.645;z= Rejection region: 1.645z>
47.2 45 2.128
6.7
42
x
zs
n
µ−−
== ≈
Reject 0
H. There is enough evidence at the 5% level of significance to reject the claim.
294 CHAPTER 7 │ HYPOTHESIS TESTING WITH ONE SAMPLE
25. 0: 5.500; : 5.500 (claim)
a
HHµµ≥<
02.33;z=− Rejection region: 2.33z<−
5.497 5.500 1.636
0.011
36
x
zs
n
µ−−
== ≈−
Fail to reject 0
H. There is not enough evidence at the 1% level of significance to support the
claim.
27. 0: $10,380 (claim); : $10,380
a
HHµµ=≠
10,240 10,380 2.54
1561
800
x
zs
n
µ−−
== ≈−
P-value = 2{Area left of 2.54z=− } = 2(0.0055) = 0.0110
Fail to reject 0
H. There is not enough evidence at the 1% level of significance to reject the claim
that the mean cost of raising a child from birth to age 2 by husband-wife families in rural areas is
$10,380.
CHAPTER 7 │ HYPOTHESIS TESTING WITH ONE SAMPLE 295
33. 0: 95; : 95 (claim)
a
HHµµ=≠
00
2.201, 2.201;tt−=− = Rejection regions: 2.201, 2.201tt<− >
94.1 95 2.038
1.53
12
x
ts
n
µ−−
== ≈−
Fail to reject 0
H. There is not enough evidence at the 5% level of significance to support the
claim.
35. 0: 0 (claim); : 0
a
HHµµ≥<
01.341;t=− Rejection region: 1.341t<−
0.45 0 1.304
1.38
16
x
ts
n
µ−−−
== ≈−
Fail to reject 0
H. There is not enough evidence at the 10% level of significance to reject the
claim.
37. 0: 48 (claim); : 48
a
HHµµ≤>
03.143;t
=
Rejection region: 3.143t>
296 CHAPTER 7 │ HYPOTHESIS TESTING WITH ONE SAMPLE
39. 0: $25 (claim); : $25
a
HHµµ=≠
00
1.740, 1.740;tt−=− = Rejection regions: 1.740, 1.740tt<− >
26.25 25 1.642
3.23
18
x
ts
n
µ−−
== ≈
Fail to reject 0
H. There is not enough evidence at the 10% level of significance to reject the
claim that the mean monthly cost of joining a health club is $25.
41. 0: $10,200 (claim); : $10,200
a
HHµµ≥<
9895.8x≈ 490.88s≈
9895.8 10,200 2.48
490.88
16
x
ts
n
µ−−
== ≈−
42. 0: $2698 (claim); : $2698
a
HHµµ=≠
2764 2698 1.085
322
28
x
ts
n
µ−−
== ≈
CHAPTER 7 │ HYPOTHESIS TESTING WITH ONE SAMPLE 297
44.
()( )
()( )
68 0.70 47.6 5
68 0.30 20.4 5 can use normal distribution
np
nq
==>
==>→
0: 0.70; : 0.70 (claim)
a
Hp Hp≥<
02.33;z=− Rejection region: 2.33z<−
l
()()
0.50 0.70 3.599
0.70 0.30
68
pp
zpq
n
−−
== ≈−
Reject
0
H. There is enough evidence at the 1% level of significance to support the claim.
l
46.
()( )
()( )
116 0.65 75.4 5
116 0.35 40.6 5 can use normal distribution
np
nq
==>
==>→
0: 0.65 (claim); : 0.65
a
Hp Hp=≠
00
2.17, 2.17;zz−=− = Rejection regions: 2.17, 2.17zz<− >
298 CHAPTER 7 │ HYPOTHESIS TESTING WITH ONE SAMPLE
48.
()( )
()( )
60 0.34 20.4 5
60 0.66 39.6 5 can use normal distribution
np
nq
==>
==>→
0: 0.34; : 0.34 (claim)
a
Hp Hp=≠
49.
()( )
()( )
50 0.24 12 5
50 0.76 38 5 can use normal distribution
np
nq
==>
==>→
0: 0.24; : 0.24 (claim)
a
Hp Hp=≠
50.
()( )
()( )
43 0.80 34.4 5
43 0.20 8.6 5 can use normal distribution
np
nq
==>
==>→
0: 0.80 (claim); : 0.80
Hp Hp≤>
51. 0: 0.16; : 0.16 (claim)
a
Hp Hp≤>
02.055;z= Rejection region: 2.055z>
l
256 0.180
1420
x
pn
== ≈
CHAPTER 7 │ HYPOTHESIS TESTING WITH ONE SAMPLE 299
52. 0: 0.02 (claim); : 0.02
a
Hp Hp=≠
00
1.96, 1.96;zz−=− = Rejection regions: 1.96, 1.96zz<− >
l
30.01
x
pn
== =
53. 2
030.144χ= 54.
22
3.565, 29.819
LR
χχ== 55. 2
063.167χ=
56. 2
01.145χ=
58. 22
0: 60 (claim); : 60
a
HHσσ≤>
2
026.119;χ= Rejection region: 226.119χ>
() ()( )
()
2
2
2
11472.7
16.963
60
ns
χσ
−
== ≈
Fail to reject 0
H. There is not enough evidence at the 2.5% level of significance to reject the
claim.
60. 0: 0.035; : 0.035 (claim)
a
HHσσ=≠
22
4.601, 32.801;
LR
χχ== Rejection regions: 22
4.601, 32.801χχ<>
() ()( )
2
2
2
22
1 15 0.026 8.278
ns
χσ
−
== ≈
300 CHAPTER 7 │ HYPOTHESIS TESTING WITH ONE SAMPLE
variance is at most 0.01.
62. 0: 3 (claim); : 3
a
HHσσ=≠
22
11.116, 48.290;
LR
χχ== Rejection regions: 22
11.116, 48.290χχ<>
() ()( )
()
2
2
2
22
1263.9
43.94
3
ns
χσ
−
== =
Fail to reject 0
H. There is not enough evidence at the 1% level of significance to reject the claim
that the standard deviation of the lengths of serving times is 3 minutes.
CHAPTER 7 QUIZ SOLUTIONS
1. (a) 0: 170 (claim); : 170
a
HHµµ≥<
(b) One-tailed because the alternative hypothesis contains <; z–test because n > 30.
(c)
01.88;z=− Rejection region: 1.88z<−
x
µ−−
2. (a) 0: 7.25 (claim); : 7.25
a
HHµµ≥<
(b) One-tailed because the alternative hypothesis contains <; t–test because n < 30, σ is
unknown, and the population is normally distributed.
CHAPTER 7 │ HYPOTHESIS TESTING WITH ONE SAMPLE 301
3. (a) 0: 0.10 (claim); : 0.10
a
Hp Hp≤>
(b) One-tailed because the alternative hypothesis contains >; z–test because 5np > and 5nq >.
(c)
01.75;z= Rejection region: 1.75z>
l
4. (a) 0: 112 (claim); : 112
a
HHσσ=≠
(b) Two-tailed because the alternative hypothesis contains ≠; 2
χ-test because the test is for a
standard deviation and the population is normally distributed.
5. (a) 0: $62,569 (claim); : $62,569
a
HHµµ=≠
(b) Two-tailed because the alternative hypothesis contains ≠; t–test because n < 30, σ is
unknown, and the population is normally distributed.
(c) Not necessary
6. (a) 0: $201 (claim); : $201
a
HHµµ=≠
(b) Two-tailed because the alternative hypothesis contains ≠; z–test because 30n≥.
(c) Not necessary