Chapter 8 Homework The claim is "the drug changes the body's temperature

CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 323

f. Fail to reject 0.H

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

decrease their times in the 40-yard dash using new strength shoes.

2a. The claim is “the drug changes the body’s temperature.”

0:0; :0 (claim)

dad

HH

µ

=≠

b. 0.05

α

= d.f. = n − 1 = 6

c. 02.447;t=± Rejection region: t < −2.447, t > 2.447

3.9 0.5771

7

d

dn

==≈

∑

()

2

22

3.9

7.29 70.9235

16

d

dn

sn

⎡⎤

⎢⎥

−⎢⎥ −

⎢⎥

⎣⎦

==≈

−

∑

324 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES

8.3 EXERCISE SOLUTIONS

3. 0:0; :0 (claim)

dad

HH

µ

≥<

0.05

α

= and d.f. = n − 1 = 13

01.771t=− (left-tailed); Rejection region: 1.771t<−

1.5 0 1.754

3.2

14

d

ts

n

µ

−−

==≈

Fail to reject 0.H

µ

5. 0:0 (claim); :0

dad

HH

µ

≤>

0.10

α

= and d.f. = n − 1 = 15

01.341 (Right-tailed);t= Rejection region: 1.341t>

6.5 0 2.725

9.54

16

d

ts

n

µ

−−

==≈

Reject

0.H

CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 325

7. 0:0 (claim); :0

dad

HH

µ

≥<

0.01

α

= and d.f. = n − 1 = 14

02.624 (Left-tailed);t=− Rejection region: 2.624t<−

2.3 0 7.423

1.2

15

d

ts

n

µ

−−−

== ≈−

Reject

0.H

µ

9. (a) The claim is a “grammar seminar will help students reduce the number of grammatical

errors.”

0:0; :0 (claim)

dad

HH

µ

≤>

(b)

03.143;t=Rejection region: 3.143t>

(c) 3.143d= and 2.035

d

s≈

(d) 3.143 0 4.086

2.035

7

d

ts

n

µ

−−

== ≈

µ

326 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES

preparation course improves the test scores of students.

11. (a) The claim is “a particular exercise program will help participants lose weight after one

month.”

0:0; :0 (claim)

dad

HH

µ

≤>

(b)

01.363;t=Rejection region: 1.363t>

(c) 3.75d= and 7.841

s≈

12. (a) The claim is “a baseball clinic will help players raise their batting average.”

0:0; :0 (claim)

dad

HH

µ

≥<

(b)

01.771;t=− Rejection region: 1.771t<−

13. (a) The claim is “soft tissue therapy and spinal manipulation help to reduce the length of time

patients suffer from headaches.”

0:0; :0 (claim)

dad

HH

µ

≤>

(b)

02.764;t=Rejection region; 2.764t>

(c) 1.225d= and 0.441

d

s≈

CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 327

14. (a) The claim is “one 600-mg dose of Vitamin C will increase muscular endurance.”

0:0; :0 (claim)

dad

HH

µ

≥<

(b)

01.761;t=− Rejection region: 1.761t<−

15. (a) The claim is “the new drug reduces systolic blood pressure.”

0:0; :0 (claim)

dad

HH

µ

≤>

(b)

01.895;t=Rejection region: 1.895t>

16. (a) The claim is “garlic can reduce the thickness of plaque buildup in arteries.”

0:0; :0 (claim)

dad

HH

µ

≤>

(b)

01.860;t=Rejection region: 1.860t>

17. (a) The claim is “the product ratings have changed from last year to this year.”

0:0; :0 (claim)

dad

HH

µ

=≠

(b)

02.365;t=± Rejection regions: 2.365, 2.365tt<− >

328 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES

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

product ratings have changed from last year to this year.

µ

(e) Fail to reject 0.H

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

scoring averages have changed.

19. Hypothesis test results:

12

:

µ

− mean of the paired difference between cholesterol (before) and cholesterol (after)

µ

P = 0.0702 > 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 new cereal lowers total blood cholesterol levels.

20. Hypothesis test results:

12

:

µ

− mean of the paired difference between time (beginning) and time (end)

01 2

:0

H

µ

µµ

−=

−≠

21. Yes; 0.0003 0.05,P≈< so you reject 0.H

CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 329

23. 1.525 and 0.542

d

ds≈− ≈

1.76 1.29

d

ss

µ

−<<−

24. 0.436 and 0.677

d

ds≈− ≈

8.4 TESTING THE DIFFERENCE BETWEEN PROPORTIONS

8.4 Try It Yourself Solutions

1a. The claim is “there is a difference between the proportion of male high school students who

smoke cigarettes and the proportion of female high school students who smoke cigarettes.

01 2 1 2

:; : (claim)

a

Hp pHp p=≠

b. 0.05

α

=

f. 12 12

12

ˆˆ

()() (0.212 0.184) (0) 4.23

11

11 0.1975 0.8025 7000 7489

pp pp

z

pq nn

−−− −−

== ≈

⎛⎞ ⎛ ⎞

⋅+

+⎜⎟

⎜⎟ ⎝⎠

⎝⎠

330 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES

2a. The claim is “the proportion of male high school students who smoke cigars is greater than the

proportion of female high school students who smoke cigars.”

01 2 1 2

:; : (claim)

a

Hp pHp p≤>

b. 0.05

α

=

c. 01.645;z= Rejection region: 1.645z>

f. 12 12

12

ˆˆ

()() (0.166 0.072) (0) 17.565

11

11 0.1174 0.8826 7000 7489

pp pp

z

pq nn

−−− −−

== ≈

⎛⎞ ⎛ ⎞

⋅+

+⎜⎟

⎜⎟ ⎝⎠

⎝⎠

8.4 EXERCISE SOLUTIONS

1. (1) The samples must be randomly selected.

(2) The samples must be independent.

CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 331

3. 01 2 1 2

:; : (claim)

a

Hp pHp p=≠

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

12

35 36 0.546

70 60

xx

pnn

++

== ≈

++

4. 01 2 1 2

:; : (claim)

a

Hp pHp p≥<

01.645;z=− Rejection region: 1.645z<−

12

471 372 0.6744

785 465

xx

pnn

++

== ≈

++

5. 01 2 1 2

: (claim); :

a

Hp p Hp p=≠

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

332 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES

12 12

12

ˆˆ

()() (0.28 0.38) (0) 1.96

11

11 (0.337)(0.663) 150 200

pp pp

z

pq nn

−−− −−

== ≈−

⎛⎞ ⎛ ⎞

+

+⎜⎟

⎜⎟ ⎝⎠

⎝⎠

Reject

0.H

7. 01 2 1 2

: (claim); :

a

Hp p Hp p≤>

01.28;z= Rejection region: 1.28z>

12

344 304 0.390

860 800

xx

pnn

++

== ≈

++

0.610q≈

8. 01 2 1 2

: (claim); :

a

Hp p Hp p=≠

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

12

29 25 0.72

45 30

xx

pnn

++

== ≈

++

CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 333

12 12

12

ˆˆ

()()

(0.644 0.833) (0) 1.786

11

11 (0.72)(0.28) 45 30

pp pp

z

pq nn

−−− −−

== ≈−

⎛⎞ ⎛ ⎞

+

+⎜⎟

⎜⎟ ⎝⎠

⎝⎠

Fail to reject 0.H

0.632q≈

12 12

12

ˆˆ

()()

(0.315 0.439) (0) 1.24

11

11 (0.368)(0.632) 54 41

pp pp

z

pq nn

−−− −−

== ≈−

⎛⎞ ⎛ ⎞

+

+⎜⎟

⎜⎟ ⎝⎠

⎝⎠

b. 01.28;z= Rejection region: 1.28z>

c. 12

12

291 249 0.9

300 300

xx

pnn

++

== ≈

++

0.1q≈

334 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES

11a. The claim is “the proportion of males who enrolled in college is less than the proportion of

females who enrolled in college.”

12

d. Reject 0.H

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

proportion of males who enrolled in college is less than the proportion of females who enrolled

in college.

12 12

12

ˆˆ

()() (0.72 0.65) (0) 2.36

11

11 (0.680)(0.320) 433 577

pp pp

z

pq nn

−−− −−

== ≈

⎛⎞ ⎛ ⎞

+

+⎜⎟

⎜⎟ ⎝⎠

⎝⎠

d. Fail to reject 0.H

e. There is not enough evidence at the 1% level of significance to reject the claim that there is no

difference in the proportion of females who have reduced the amount they spend on eating out

and the proportion of males who have reduced the amount they spend on eating out.

CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 335

e. There is enough evidence at the 5% level of significance to reject the claim that the proportion

of subject who are pain-free is the same for the two groups.

d. Reject 0.H

e. There is enough evidence at the 10% level of significance to support the claim that the

proportion of subjects who are free of nausea is greater for subjects who took the drug than for

subjects who took placebo.”

16a. The claim is “the proportion of motorcyclists who wear a helmet in the Northeast is less than

the proportion of motorcyclists who wear helmet in the Midwest.”

01 2 1 2

:; : (claim)

a

Hp pHp p≥<

b. 01.28;z=− Rejection region: 1.28z<−

336 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES

e. There is enough evidence at the 10% level of significance to support the claim that the

proportion of motorcyclists who wear a helmet in the Northeast is less than the proportion of

motorcyclists who wear helmet in the Midwest.

18a. The claim is “the proportion of adults who use the Internet is greater for adults who live in an

urban area than for adults who live in a rural area.”

01 2 1 2

:; : (claim)

a

Hp pHp p≤>

b. 01.28;z= Rejection region: 1.28z>

c. 12

12

359 221 0.725

485 315

xx

pnn

++

== ≈

++

0.275q≈

19. 01 2 1 2

: (claim); :

a

Hp p Hp p=≠

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

12

240 204 0.555

400 400

xx

pnn

++

== ≈

++

CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 337

Reject 0.H

There is enough evidence at the 5% level of significance to reject the claim that the proportion of

customers who wait 20 minutes or less is the same at the Fairfax North and Fairfax South offices.

Reject 0.H

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

customers who wait 20 minutes or less is greater at the Staunton office that it is at the Fairfax

South office.

21. 01 2 1 2

:; : (claim)

a

Hp pHp p≥<

01.28;z=− Rejection region: 1.28z<−

Reject 0.H

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

of customers who wait 20 minutes or less at the Roanoke office is less than the proportion of

customers who wait 20 minutes or less at the Staunton office.