CHAPTER
Hypothesis Testing with Two Samples
303
8
8.1 TESTING THE DIFFERENCE BETWEEN MEANS
(LARGE INDEPENDENT SAMPLES)
8.1 Try It Yourself Solutions
Note: Answers may differ due to rounding.
2. The claim is “there is a difference in the mean annual wages for forensic science technicians
working for local and state governments.”
a. 01 2 1 2
: ; : (claim)
a
HH
µ
µµµ
=≠
b,
α
= 0.10
c. 01.645;z= Rejection regions: z > 1.645, z < −1.645
d. 12 1 2
22 2 2
12
12
()( )
(53,000 51,910) (0) 1.667
(6200) (5575)
100 100
xx
z
ss
nn
µ
µ
−−− −−
== ≈
+
+
3a. 12 1 2
22 2 2
12
12
()( )
(274 271) (0) 1.36
(22) (18)
150 200
xx
z
ss
nn
µ
µ
−−− −−
==≈
+
+
304 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES
8.1 EXERCISE SOLUTIONS
2. State the hypotheses and identify the claim. Specify the level of significance. Find the critical
values(s) and identify the rejection region(s). Find the standardized test statistic. Make a decision
and interpret it in the context of the claim.
3. Use P-values.
5. Independent because different students were samples.
6. Dependent because the same students were sampled.
7. Dependent because the same football players were sampled.
11. Dependent because the same tire sets were sampled.
12. Dependent because the same people were sampled.
CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 305
14. 01 2 1 2
:; : (claim)
a
HH
µ
µµµ
≤>
Rejection region: 01.28z>
(a)
12
500 495 5xx−= − =
µ
15. 01 2 1 2
:; : (claim)
a
HH
µ
µµµ
≥<
Rejection region: 01.645z<−
(a)
12
2435 2432 3xx−= − =
µ
16. 01 2 1 2
: (claim); :
a
HH
µ
µµµ
≤>
Rejection region: 01.88z>
(a)
12
5004 4895 109xx−= − =
µ
17. 01 2 1 2
:; : (claim)
a
HH
µ
µµµ
≤>
02.33;z= Rejection region: 2.33z>
12 1 2
22 2 2
12
()( )
(5.2 5.5) (0) 5.207
(0.2) (0.3)
45 37
xx
z
ss
nn
µ
µ
−−− −−
==≈−
+
+
306 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES
µ
19. Because 2.96 1.96z≈> and 0.0031 0.05,P≈<reject 0.H
20. Because 1.94 2.33z≈< and 0.0261 0.01,P≈> reject 0.H
21. The claim is “the mean braking distances are different for the two types of tires.”
(a)
01 2 1 2
:; : (claim)
a
HH
µ
µµµ
=≠
(b)
01.645;z=± Rejection regions: 1.645, 1.645zz<− >
µ
22. The claim is “the mean braking distance for Type C is greater than the mean braking distance for
Type D.”
(a)
01 2 1 2
:; : (claim)
a
HH
µ
µµµ
≤>
(b)
01.28;z= Rejection region: 1.28z>
µ
23. The claim is “Region A’s average wind speed is greater than Region B’s.”
(a)
01 2 1 2
:; : (claim)
a
HH
µ
µµµ
≤>
(b)
01.645;z> Rejection region: 1.645z>
µ
CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 307
(d) Fail to reject 0.H
(e) There is not enough evidence at the 5% level of significance to conclude that Region A’s
average wind speed is greater than Region B’s.
µ
25. The claim is “male and female high school students have equal ACT scores.”
(a)
01 2 1 2
: (claim); :
a
HH
µ
µµµ
=≠
(b)
02.575;z=± Rejection regions: 2.575, 2.575zz<− >
µ
26. The claim is “high school students in a college preparation program have higher ACT scores than
those in a general program.”
(a)
01 2 1 2
:; : (claim)
a
HH
µ
µµµ
≤>
(b)
01.28;z= Rejection region: 1.28z>
(c) 12 1 2
22 2 2
12
12
()( )
(22.2 20.0) (0) 2.07
(4.8) (5.4)
49 4
xx
z
ss
nn
µ
µ
−−− −−
==≈
+
+
µ
308 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES
(c) 12 1 2
22 2 2
12
12
()( )
(240,993 249,237) (0) 1.30
(25,875) (27,110)
35 35
xx
z
ss
nn
µ
µ
−−− −−
== ≈−
+
+
(d) Fail to reject 0.H
(e) There is not enough evidence at the 10% level of significance to reject the claim that the
average home sales price in Dallas, Texas is the same as in Austin, Texas.
µ
µ
29. The claim is “the average home sales price in Dallas, Texas is the same as in Austin, Texas.”
(a)
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
12
()( )
(247,245 239,150) (0) 1.86
(23,740) (20,690)
50 50
xx
z
ss
nn
µ
µ
−−− −−
== ≈
+
+
(d) Reject
0.H
(e) There is enough evidence at the 10% level of significance to reject the claim that the average
home sales price in Dallas, Texas is the same as in Austin, Texas. The new samples do lead to
a different conclusion.
µ
µ
CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 309
(d) Reject
0.H
(e) There is enough evidence at the 5% level of significance to support the claim that households
in the United States headed by people under the age of 25 spend less on food away from
home than households headed by people ages 55−64.
The new samples do lead to a different conclusion.
µ
32. The claim is “middle school boys spent less time studying in 1981 than middle school boys do
today.”
(a)
01 2 1 2
:;: (claim)
a
HH
µ
µµµ
≥<
(b)
01.88;z=− Rejection region: 1.88z<−
(c)
111
32.523, 4.477, 35xsn≈≈=
222
47.989, 4.651, 35xsn≈≈=
12 1 2
22 2 2
12
12
()( )
(32.523 47.989) (0) 14.17
(4.477) (4.651)
35 35
xx
z
ss
nn
µ
µ
−−− −−
== ≈−
+
+
(d) Reject
0.H
(e) There is enough evidence at the 3% level of significance to support the claim that middle
school boys spent less time studying In 1981 than middle school boys do today.
µ
µ
310 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES
(e) There is enough evidence at the 1% level of significance to reject the claim that there is no
difference in the mean washer diameter manufactured by two different methods.
µ
35. They are equivalent through algebraic manipulation of the equation.
12 12
0
µ
µµµ
=→−=
36. They are equivalent through algebraic manipulation of the inequality.
12 12
0
µ
µµµ
≥→−≥
37. Hypothesis test results:
1:
µ
mean of population 1 (Std. Dev. = 5.4)
2:
µ
mean of population 2 (Std. Dev. = 7.5)
µ
µ
µ
P = 0.0031 < 0.01, so reject 0.H
There is enough evidence at the 1% level of significance to support the claim.
38. Hypothesis test results:
1:
µ
mean of population 1 (Std. Dev. = 20.8)
2:
µ
mean of population 2 (Std. Dev. = 24.6)
12
:
µ
µ
− mean difference
µ
CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 311
Difference 1
n 2
n Sample Mean
12
µ
µ
− 80 80 3.6
39. Hypothesis test results:
1:
µ
mean of population 1 (Std. Dev. = 0.92)
2:
µ
mean of population 2 (Std. Dev. = 0.73)
12
:
µ
µ
− mean difference
01 2
12
:0
:0
a
H
H
µ
µ
µµ
−=
−<
µ
P-value = 0.0492 < 0.05, so reject 0.H
There is enough evidence at the 5% level of significance to reject the claim.
40. Hypothesis test results:
1:
µ
mean of population 1 (Std. Dev. =193.8)
2:
µ
mean of population 2 (Std. Dev. = 129.25)
µ
µ
µ
312 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES
41. 01 2 1 2
:9(claim); :9
a
HH
µ
µµµ
−=− −≠−
02.575;z=± Rejection region: 2.575, 2.575zz<− >
12 1 2
22 2 2
12
12
()( )
(11.5 20) ( 9) 0.528
(3.8) (6.7)
70 65
xx
z
ss
nn
µ
µ
−−− −−−
==≈
+
+
Fail to reject 0.H There is not enough evidence at the 1% level of significance to reject the claim
that children spend 9 hours a week more in day care or preschool today than in 1981.
µ
µ
43. 01 2 1 2
: 10,000; : 10,000 (claim)
a
HH
µ
µµµ
−≤ −>
01.645;z= Rejection region: 1.645z>
12 1 2
22 2 2
12
12
()( )
(94,980 80,830) (10,000) 2.05
(8795) (9250)
42 38
xx
z
ss
nn
µ
µ
−−− −−
== ≈
+
+
Reject
0.H There is enough evidence at the 5% level of significance to support the claim that the
mean annual salaries of microbiologists in California and Maryland is more than $10,000.
44. 01 2 1 2
: 15,000; : 15,000 (claim)
a
HH
µ
µµµ
−≤ −>
01.28;z= Rejection region: 1.28z>
12 1 2
22 2 2
12
()( )
(88,540 72,870) (15,000) 0.33
(8225) (7640)
30 32
xx
z
ss
nn
µ
µ
−−− −−
== ≈
+
+
CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 313
46.
22 22
12 12
12 1 2 12
12 12
22 22
12
12
12
() ()
(1.2) (1.5) (1.2) (1.5)
(10.3 8.5) 1.96 (10.3 8.5) 1.96
140 127 140 127
1.8 1.96 0.028 1.8 1.96 0.028
1.5 2.1
cc
ss ss
xx z xx z
nn nn
µµ
µµ
µµ
µµ
−− +<−<−+ +
−− + <−< −+ +
−− <−< +
<− <
µ
µ
48. 01 2 1 2
: 0; : 0 (claim)
a
HH
µ
µµµ
−≤ −>
01.645;z= Rejection region: 1.645z<
12 1 2
22 2 2
12
12
()( )
(10.3 8.5) (0) 10.76
(1.2) (1.5)
140 127
xx
z
ss
nn
µ
µ
−−− −−
==≈
+
+
Reject
0.H There is enough evidence at the 5% level of significance to support the claim. You
should recommend using Irinotecan because the average number of months with no reported
cancer related pain was significantly higher with fluorouracil.
µ
µ
µ
µ
50. 01 2 1 2
: 0; : 0 (claim)
a
HH
µ
µµµ
−≤ −>
The 95% CI for 12
µ
µ
−in Exercise 46 contained only values greater than zero and, as found in
Exercise 48, there was enough evidence at the 5% level of significance to support the claim. If the
µ
µ
314 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES
8.2 TESTING THE DIFFERENCE BETWEEN MEANS
(SMALL INDEPENDENT SAMPLES)
8.2 Try It Yourself Solutions
1a. The claim is “there is a difference in the mean annual earnings based on level of education.”
01 2 1 2
:; : (claim)
a
HH
µ
µµµ
=≠
b. 0.01;
α
= d.f. =
{
}
12
min{ 1, 1} min 15 1, 12 1 11nn−−= −−=
c. 03.106;t=± Rejection regions: 3.106, 3.106tt<− >
µ
e. Reject 0.H
f. There is evidence that “there is a difference in the mean annual earnings based on level of
education.”
2a. The claim is “the watts usage of a manufacturer’s 17-inch flat panel monitors is less than that of
its leading competitor.”
01 2 1 2
:; : (claim)
a
HH
µ
µµµ
≥<
µ
8.2 EXERCISE SOLUTIONS
1. (1) The samples must be randomly selected.
(2) The samples must be independent
CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 315
2. State hypotheses and identify the claim. Specify the level of significance. Determine the degrees
of freedom. Find the critical value(s) and identify the rejection region(s). Find the standardized
test statistic. Make a decision and interpret it in the context of the original claim.
5. (a) d.f. = 12
216nn+−= 6. (a) d.f. = 12
239nn+−=
01.746t=− 02.575t=
(b) d.f. = 12
min{ 1, 1} 6nn−−= (b) d.f. =
12
min{ 1, 1} 18nn−−=
01.943t=−
02.878t=±
01.895t=
01.638t=−
9. 01 2 1 2
: (claim); :
a
HH
µ
µµµ
=≠
d.f. =
12
227nn+−=
Rejection regions: 2.771, 2.771tt<− >
(a)
12
33.7 33.5 1.8xx−= − =−
µ
10. 01 2 1 2
:; : (claim)
a
HH
µ
µµµ
≥<
d.f. =
12
218nn+−=
Rejection region: 1.33t<−
(a)
12
0.345 0.515 0.17xx−=−=−
µ
316 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES
11. 01 2 1 2
: (claim); :
a
HH
µ
µµµ
≤>
d.f. = 12
min{ 1, 1} 9nn−−=
Rejection region: 1.833t>
(a)
12
2410 2305 105xx−= − =
µ
12. 01 2 1 2
:; : (claim)
a
HH
µ
µµµ
≤>
d.f. = 12
min{ 1, 1} 14nn−−=
Rejection region: 2.624t>
(a)
12
52 50 2xx−=−=
µ
13. (a) The claim is “the mean annual costs of routine veterinarian visits for dogs and cats are the
same.”
01 2 1 2
: (claim); :
a
HH
µ
µµµ
=≠
(b) d.f. = 12
min{ 1, 1} 6nn−−=
01.943;t=± Rejection regions: 1.943, 1.943tt<− >
µ
14. (a) The claim is “athletes have a greater mean maximal oxygen consumption than non-athletes.”
01 2 1 2
:; : (claim)
a
HH
µ
µµµ
≤>
(b) d.f. = 12
242nn+−=
01.645;t= Rejection region: 1.645t>
CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 317
15. (a) The claim is “the mean bumper repair cost is less for mini cars than for midsize cars.”
01 2 1 2
:; : (claim)
a
HH
µ
µµµ
≥<
(b) d.f. = 12
220nn+−=
01.325;t=− Rejection region: 1.325t<−
µ
16. (a) The claim is “the mean footwell instrusions for small pickups and small SUVs are equal.”
01 2 1 2
: (claim); :
a
HH
µ
µµµ
=≠
(b) d.f. = 12
218nn+−=
02.878;t=± Rejection regions: 2.878, 2.878tt<− >
(c) 12 1 2
22 2 2
12 2
12 1 2
()( ) (11.18 9.52) (0) 0.87
( 1) ( 1) (7 1)(4.53) (13 1)(3.84) 1 1
11
7132 7 13
2
xx
t
ns n s
nn n n
µ
µ
−−− −−
== ≈
−+− − +− +
++−
+−
(d) Fail to reject 0.H
(e) The evidence supports the claim at the 1% level of significance that the mean footwell
intrusions for small pickups and small SUVs are equal.
µ
318 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES
(d) Reject 0.H
(e) There is enough evidence at the 5% level of significance to support the claim that the mean
household income is greater in Allegheny County than it is in Erie County.
(c) 12
22 2 2
12
12
()(0)
(49,800 44,400) (0) 2.38
(4200) (8600)
17 18
xx
t
ss
nn
−− −−
== ≈
+
+
(d) Fail to reject 0.H
(e) There is not enough evidence at the 1% level of significance to support the claim that the
mean household income is greater in Hillsborough County than it is in Polk County.
µ
(d) Fail to reject 0.H
(e) There is not enough evidence at the 1% level of significance to support the claim that the new
treatment makes a difference in the tensile strength of steel bars.
20. (a) The claim is “the experimental method produces steel with greater mean tensile strength.”
01 2 1 2
:; : (claim)
a
HH
µ
µµµ
≤>
(b) d.f. = 12
min{ 1, 1} 13nn−−=
µ
CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 319
(e) There is enough evidence at the 10% level of significance to support the claim that the
experimental method produces steel with greater mean tensile strength and to recommend
using the experimental method.
21. (a) The claim is “the new method of teaching reading produces higher reading test scores than
the old method”
01 2 1 2
:; : (claim)
a
HH
µ
µµµ
≥<
(b) d.f. = 12
242nn+−=
(d) Reject 0.H
(e) There is enough evidence at the 10% level of significance to support the claim that the new
method of teaching reading produces higher reading test scores than the old method.
01.645;t=− Rejection region: 1.645t<−
(c)
111
79.091, 6.900, 22xsn≈≈=
22 2
83, 7.645, 19xs n≈≈ =
()( ) (79.091 83) (0) 1.721
xx
µ
µ
−−− −−
320 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES
23. Hypothesis test results:
1:
µ
mean of population 1
2:
µ
mean of population 2
12
:
µ
µ
− mean difference
µ
µ
P = 0.6789 > 0.10, so fail to reject 0.H
There is not enough evidence at the 10% level of significance to support the claim.
24. Hypothesis test results:
1:
µ
mean of population 1
2:
µ
mean of population 2
12
:
µ
µ
− mean difference
µ
µ
CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES 321
25. Hypothesis test results:
1:
µ
mean of population 1
2:
µ
mean of population 2
12
:
µ
µ
− mean difference
µ
µ
P = 0.0714 > 0.05, so fail to reject 0.H
There is enough evidence at the 5% level of significance to support the claim.
26. Hypothesis test results:
1:
µ
mean of population 1
2:
µ
mean of population 2
12
:
µ
µ
− mean difference
µ
µ
P = 0.0317 < 0.10, so reject 0.H
There is not enough evidence at the 10% level of significance to reject the claim.
27.
22 22
12 2
12
(1) ( 1) (21 1)(166) (11 1)(204) 179.56
221112
ns n s
nn
σ
−+− −+−
== ≈
+− +−
322 CHAPTER 8 │ HYPOTHESIS TESTING WITH TWO SAMPLES
28.
22 22
12 2
12
(1) ( 1) (26 1)(8.5) (24 1)(11.5) 10.050
2 26242
ns n s
nn
σ
−+− −+−
== ≈
+− +−
12 12
29.
22 22
12
12
12
12 12
(6) (12)
( ) (267 244) 2.132 95
23 12.21 10.78 35.21 11 35
c
ss
xx t
nn
µµ µµ
−± +→ − ± +
→± → <−< →<−<
8.3 TESTING THE DIFFERENCE BETWEEN MEANS
(DEPENDENT SAMPLES)
8.3 Try It Yourself Solutions
1a. The claim is “athletes can decrease their times in the 40-yard dash.”
0:0; :0 (claim)
dad
HH
µ
µ
≤>
b. 0.05
α
= d.f. = n − 1 = 11
c. 01.796;t= Rejection region: t > 1.796
d.
Before After d 2
d
4.85
4.90
5.08
4.72
4.62
4.78
4.90
5.05
4.65
4.64
0.07
0.00
0.03
0.07
−0.02
0.0049
0.0000
0.0009
0.0049
0.0004
0.28 0.0233
12
d
d
n
==≈
∑