Chapter 8 Homework There is not enough evidence at the 5% level of significance

338 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES

12 12

12

ˆˆ

()() (0.56 0.60) (0) 1.15

11

11 (0.58)(0.42) 400 400

pp pp

z

pq nn

−−− −−

== ≈−

⎛⎞ ⎛ ⎞

+

+⎜⎟

⎜⎟ ⎝⎠

⎝⎠

24. Yes; When 0.10,

α

= the rejection regions become z < −1.645 and z> 1.645. Because −1.645 <

−1.15 < 1.645, you still fail to reject 0.H

25. Hypothesis test results:

1

p:proportion of successes for population 1

2

p:proportion of successes for population 2

P = 0.2518 > 0.05, so fail to reject 0

H.

There is not enough evidence at the 5% level of significance to support the claim that the

proportion of men ages 18−24 living in their parents’ homes was greater in 2000 than in 2009.

26. Hypothesis test results:

1

p:proportion of successes for population 1

2

p:proportion of successes for population 2

12

pp:− difference in proportions

CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 339

27. Hypothesis test results:

1

p:proportion of successes for population 1

2

p:proportion of successes for population 2

P < 0.0001 < 0.01, so reject 0

H.

There is enough evidence at the 1% level of significance to reject the claim that the proportion of

18 to 24-year olds living in their parents’ homes in 2000 was the same for men and women.

28. Hypothesis test results:

1

p:proportion of successes for population 1

2

p:proportion of successes for population 2

P = 0.0001 < 0.01, so reject 0

H.

There is enough evidence at the 10% level of significance to reject the claim that the proportion

of 18 to 24-year olds living in their parents’ homes in 2009 was the same for men and women.

340 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES

CHAPTER 8 REVIEW EXERCISE SOLUTIONS

1. Dependent because the same cities were sampled.

2. Dependent because the same runners were sampled.

3. 01 2 1 2

: (claim); :

a

HH

µ

µµµ

≥<

µ

4. 01 2 1 2

: (claim); :

a

HH

µ

µµµ

=≠

02.575;z=± Rejection region: 2.575, 2.575zz<− >

µ

0.HThere is enough evidence at the 1% level of significance to reject the claim.

5. 01 2 1 2

:; : (claim)

a

HH

µ

µµµ

≥<

01.28;z=− Rejection regions: 1.28z<−

µ

CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 341

6. 01 2 1 2

:; : (claim)

a

HH

µ

µµµ

=≠

01.96;z=± Rejection regions: 1.96, 1.96zz<− >

µ

7. (a) 01 2 1 2

:; : (claim)

a

HH

µ

µµµ

≥<

(b)

01.645;z=− Rejection region: 1.645z<−

()( )

(1010 1180) (0) 9.20

xx

µ

−−− −−

8. (a) The claim is “the mean annual salary of civilian federal employees in California is the same

as those in Illinois.”

01 2 1 2

: (claim); :

a

HH

µ

µµµ

=≠

(b)

01.645;z=− Rejection regions: 1.645, 1.645zz<− >

(c) 12 1 2

22 2 2

12

()( )

(66,210 67,390) (0) 1.81

(6385) (5998)

180 180

xx

z

ss

nn

µ

−−− −−

== ≈−

+

(d) Reject

0.H

(e) There is enough evidence at the 10% level of significance to reject the claim that the mean

annual salary of civilian federal employees in California is the same as those in Illinois.

9. Yes; The new rejection region is z < −2.33, which contains z = −9.20, so you still reject 0.H

µ

342 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES

Reject 0.H There is not enough evidence at the 5% level of significance to support the claim.

µ

13. 01 2 1 2

: (claim); :

a

HH

µ

µµµ

≤>

d.f. = 12

min{ 1, 1} 24nn−−=

01.711;t= Rejection regions: 1.711t>

12 1 2

22 2 2

12

()( )

(183.5 184.7) (0) 1.460

(1.3) (3.9)

25 25

xx

t

ss

nn

µ

−−− −−

==≈−

+

Fail to reject 0.HThere is not enough evidence at the 5% level of significance to reject the claim.

µ

15. 01 2 1 2

:; : (claim)

a

HH

µ

µµµ

=≠

d.f. = 12

210nn+−=

03.169;t=± Rejection regions: 3.169, 3.169tt<− >

µ

CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 343

16. 01 2 1 2

: (claim); :

a

HH

µ

µµµ

≥<

d.f. = 12

min{ 1, 1} 5nn−−=

µ

17. (a) The claim is “third graders taught with the directed reading activities scored higher than those

taught without the activities.”

01 2 1 2

:; : (claim)

a

HH

µ

µµµ

≤>

(b) d.f. = 12

242nn+−=

01.645;t= Rejection region: 1.645t>

(c)

111

51.476, 11.007, 21xsn≈≈=

µ

18. (a) The claim is “there is no difference between the mean household incomes of two

neighborhoods.”

01 2 1 2

: (claim); :

a

HH

µ

µµµ

=≠

(b) d.f. = 12

220nn+−=

02.845;t=± Rejection regions: 2.845, 2.845tt<− >

µ

344 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES

19. 0:0 (claim); :0

dad

HH

µ

=≠

0.01

α

= and d.f. = n − 1 = 15

0.H

20. 0:0; :0 (claim)

dad

HH

µ

≥<

0.10

α

= and d.f. = n − 1 = 24

01.318t=− (Left-tailed test); Rejection regions: 1.318t<−

21. 0:0 (claim); :0

dad

HH

µ

≤>

0.10

α

= and d.f. = n − 1 = 32

01.282t= (Right-tailed test); Rejection region: 1.282t>

10.3 0 3.253

18.19

33

d

ts

n

µ

−−

== ≈

Reject

0.H

µ

CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 345

(c)

d = 5 and 8.743

d

s≈

(d) 50 1.808

8.743

10

d

ts

n

µ

−−

==≈

(e) Reject

0.H

(f) There is enough evidence at the 10% level of significance to support the claim that the man’s

systolic blood pressure decreased.

25. 01 2 1 2

: (claim); :

a

Hp p Hp p=≠

01.96;z=± Rejection regions: 1.96, 1.96zz<− >

12

425 410 0.522

840 760

xx

pnn

++

== ≈

++

0.478q≈

26. 01 2 1 2

: (claim): :

a

Hp p Hp p≤>

02.33;z= Rejection region: 2.33z>

346 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES

27. 01 2 1 2

:; : (claim)

a

Hp pHp p≤>

01.28;z= Rejection region: 1.28z>

12

261 207 0.450

556 483

xx

pnn

++

== ≈

++

0.550q≈

112 2

250.2 5, 305.8 5, 217.35 5, and 265.65 5np nq n p n q≈>≈>≈> ≈>.

28. 01 2 1 2

:; : (claim)

a

Hp pHp p≥<

01.645;z=− Rejection region: 1.645z<−

12

86 107 0.092

900 1200

xx

pnn

++

== ≈

++

0.908q≈

11 2 2

82.8 5, 817.2 5, 110.4 5, and 1089.6 5np nq n p n q≈> ≈ > ≈ > ≈ >.

Can use normal sampling distribution.

CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 347

(c) 12

12

468 472 0.488

900 1027

xx

pnn

++

== ≈

++

0.512q≈

30. (a) The claim is “the proportion of U.S. adults who believe it is likely that life exists on other

planets is less now than in 2007.”

01 2 1 2

:; : (claim)

a

Hp pHp p≤>

(b)

01.645;z= Rejection region: 1.645z>

(c) 12

12

570 530 0.55

1000 1000

xx

pnn

++

== ≈

++

31. Yes; When 0.05,

α

= the rejection regions become z < −1.96 and z > 1.96. Because 2.65 > 1.96,

you still reject 0.H

32. No; When 0.01,

α

= the rejection region becomes z > 2.33. Because 1.80 < 2.33, you fail to reject

0.H

CHAPTER 8 QUIZ SOLUTIONS

1. (a) 01 2 1 2

: ; : (claim)

a

HH

µ

µµµ

≤>

(b) One-tailed because

H contains >; z-test because 1

n and 2

n are each greater than 30.

348 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES

(e) Fail to reject 0.H

(f) There is not enough evidence at the 5% level of significance to support the claim that the

mean score on the science assessment for male high school students was higher than for the

female high school students.

(d) 12 1 2

22 22

12 2

12 1 2

()( ) (153 149) (0) 0.341

(1) ( 1) 1 1 (13 1)(32) (15 1)(30) 1 1

2 13 15 2 13 15

xx

t

ns n s

nn n n

µ

−−− −−

== ≈

−+− −+−

⋅+ ⋅+

+− +−

(e) Fail to reject 0.H

(f) There is not enough evidence at the 1% level of significance to reject the claim that the mean

score on the science assessment test are the same for fourth grade boys and girls.

(e) Fail to reject 0.H

(f) There is not enough evidence at the 10% level of significance to reject the claim that the

proportion of U.S. adults who are worried that their family will become a victim of terrorism

has not changed.

4. (a) 0:0; :0 (claim)

dad

HH

µ

≥<

CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 349

CUMULATIVE REVIEW, CHAPTERS 6−8

1. (a) ˆˆ

0.13, 0.87pq==

ˆˆ (0.13)(0.87)

ˆ0.13 1.96 0.13 0.021 (0.109, 0.151)

1000

c

pq

pz n

±=± ≈±=

(b)

0: 0.10; : 0.10 (claim)

a

Hp Hp≤>

01.645;z= Rejection region: 1.645z>

2. 0:0; :0 (claim)

dad

HH

µ

≥<

0.10

α

= d.f. = n − 1 = 7

01.415;t=− Rejection region: t < −1.415

1.575d=− and 0.803

d

s≈

3. 3.4

26.97 1.96 26.97 1.03 (25.94, 28.00);

42

c

s

xz n

±=± ≈±= z–distribution

4. 1.63

3.46 1.753 3.46 0.71 (2.75, 4.17);

16

c

s

xt n

±=± ≈±= t-distribution

350 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES

6. 0.62

8.21 2.365 8.21 0.52 (7.69, 8.73);

8

c

s

xt n

±=± ≈±= t–distribution

7. 01 2 1 2

:; : (claim)

a

HH

µ

µµµ

≤>

01.28;z= Rejection region: 1.28z>

8. 0:33H

µ

≥ 9. 0:0.19 (claim)Hp≥

:33

a

H

µ

< (claim) :0.19

a

Hp<

10. 0:0.63 (claim)H

σ

= 11. 0:2.28H

µ

=

:0.63

a

H

σ

≠ :2.28 (claim)

a

H

µ

≠

12. (a)

22 2 2

22

( 1) ( 1) (26 1)(3.1) (26 1)(3.1)

, , (5.1, 22.8)

46.928 10.520

RL

ns ns

χχ

⎛⎞

−− − −

=≈

⎜⎟

⎝⎠

(b)

()

5.1, 22.8 (2.3, 4.8)≈

(c)

0: 2.5; : 2.5 (claim)

a

HH

σσ

≥<

13. 01 2 1 2

:; : (claim)

a

HH

µ

µµµ

=≠

d.f. = 12

22618242nn+−=+−=

01.96;t=± Rejection regions: 1.960, 1.960tt<− >

CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 351

Reject 0.H There is enough evidence at the 5% level of significance to support the claim that the

mean SAT scores for male athletes and male non-athletes at a college are different.

µ

15. 01 2 1 2

: (claim); :

a

Hp p Hp p=≠

12

195 204 0.621

319 323

xx

pnn

++

== ≈

++

16. (a) 1.6

42 1.96 42 0.496 (41.5, 42.5)

40

s

xz

n

±=± ≈± ≈

(b)

0: 45 (claim); : 45

a

HH

µ

≥<

01.96;z=− Rejection region: z < −1.96

µ