CHAPTER 10 │ CHI-SQUARE TESTS AND THE
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23a−b. Contingency table results:
Rows: Expected income
Columns: None
Cell format
Count
Expected count
Did not
consider it Total
Less than
$35,000
25
25.85
94
35,000 to
$50,000
16
19.52
71
$50,000 to
$100,000
48
48.68
177
Greater than
$100,000
32
26.95
98
Total 121 440
422 CHAPTER 10 │ CHI-SQUARE TESTS AND THE
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16−20 21−30 31−40 41−50
Male 45
44.8
170
165.9
90
92.09
72
73.84
Female 9
9.2
30
34.07
21
18.91
17
15.16
Total 54 200 111 89
c. Fail to reject 0.H There is not enough evidence at the 5% level of significance to conclude that
age is related to gender in fatal alcohol-related accidents of passenger vehicle drivers with blood
alcohol concentrations greater than or equal to 0.08.
25. The claim is “the proportions of motor vehicle crash deaths involving males or females are the
same for each age group.”
0:H The proportions are equal. (claim)
:
a
H At least one of the proportions is different from the others.
d.f. = (r − 1)(c − 1) = 7
2
0
χ
= 14.067; Rejection region: 2
χ
> 14.067
O E O
−
E OE
2
()− OE
E
2
()
−
123 121.632 1.368 1.871424 0.01539
97 89.965 7.035 49.491225 0.55012
82 79.169 2.831 8.014561 0.10123
82 82.048 −0.048 0.002304 0.00003
56 56.138 −0.138 0.109044 0.00034
28.134
χ
≈
26. The claim is “the proportions of the results for drug and placebo treatments or the same.”
0:H The proportions are equal. (claim)
:
a
H At least one of proportion is different from the others.
d.f. = (r − 1)(c − 1) = 1
2
0
χ
= 2.706; Rejection region: 2
χ
> 2.706
O E O
−
E OE
2
()− OE
E
2
()
−
39 31.66 7.34 53.8756 1.7017
27. Right-tailed
28. Answers will vary. Sample answer: Both tests are very similar, but the chi-square test for
independence tests whether the occurrence of one variable affects the probability of the
29.
Educational attainment
Status
Not a
high school
graduate
High
school
graduate
Some
college,
no degree
Associate’s,
bachelor’s,
or advanced
degree
Employed 0.055 0.183 0.114 0.290
Unemployed 0.006 0.011 0.005 0.007
31. Several of the expected frequencies are less than 5.
32.
Educational attainment
Status
Not a
high school
graduate
High
school
graduate
Some
college,
no degree
Associate’s,
bachelor’s,
or advanced
degree
Employed 0.086 0.285 0.177 0.452
34. 16.3%
35.
Educational attainment
Status
Not a
high school
graduate
High
school
graduate
Some
college,
no degree
Associate’s,
bachelor’s,
or advanced
degree
Employed 0.411 0.587 0.660 0.759
Unemployed 0.046 0.036 0.030 0.019
Not in the
labor force 0.544 0.377 0.311 0.223
CHAPTER 10 │ CHI-SQUARE TESTS AND THE
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10.3 COMPARING TWO VARIANCES
10.3 Try It Yourself Solutions
1a. 0.05
α
= b. 0
F = 2.45
2a. 0.01
α
= b. 0
F = 18.31
e.
2
1
2
2
180 3.21
56
s
Fs
== ≈
f. Reject 0.H
g. There is enough evidence at the 1% level of significance to support the researcher’s claim that a
specially treated intravenous solution decreases the variance of the time required for nutrients to
enter the bloodstream.
10.3 EXERCISE SOLUTIONS
1. Specify the level of significance .
α
Determine the degrees of freedom for the numerator and
denominator. Use Table 7 in Appendix B to find the critical value F.
426 CHAPTER 10 │ CHI-SQUARE TESTS AND THE
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(5) For all F-distributions, the mean value of F is approximately equal to 1.
3. (1) The samples must be randomly selected, (2) the samples must be independent, and (3) each
population must have a normal distribution.
11. 01.80F= 12. 02.42F=
13. 22 22
01 2 1 2
:; :
a
HH
σσ σσ
≤> (claim)
N
d.f. 4=
D
d.f. 5=
03.52;F= Rejection region: F > 3.52
2
1
2
2
773 1.010
765
s
Fs
== ≈
Fail to reject 0
H. There is not enough evidence at the 10% level of significance to support the
claim.
15. 22 22
01 2 1 2
: (claim); :
a
HH
σσ σσ
≤>
N
d.f. 10=
D
d.f. 9=
05.26;F= Rejection region: F > 5.26
CHAPTER 10 │ CHI-SQUARE TESTS AND THE
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16. 22 22
01 2 1 2
:; :
a
HH
σσ σσ
=≠ (claim)
N
d.f. 30=
D
d.f. 27=
17. 22 22
01 2 1 2
: (claim); :
a
HH
σσ σσ
=≠
N
d.f. 12=
D
d.f. 19=
03.76;F= Rejection region: F > 3.76
2
1
2
2
9.8 3.920
2.5
s
Fs
== ≈
Reject
0
H. There is enough evidence at the 1% level of significance to reject the claim.
19. Population 1: Company B
Population 2: Company A
a. The claim is “the variance of the life of Company A’s appliances is less than the variance of the
life of Company B’s appliances.”
22 22
01 2 1 2
:; :
a
HH
σσ σσ
≤> (claim)
b. N
d.f. 24=
D
d.f. 19=
02.11;F= Rejection region: F > 2.11
428 CHAPTER 10 │ CHI-SQUARE TESTS AND THE
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20. Population 1: Competitor
Population 2: Auto Manufacturer
a. The claim is “the variance of the life of the fuel consumption for the company’s hybrid vehicles is
less than that of the competitor’s hybrid vehicles.”
22 22
01 2 1 2
:; :
a
HH
σσ σσ
≤> (claim)
21. Population 1: Company B
Population 2: Company A
a. The claim is “the variance of the prices differ between the two companies.”
22 22
01 2 1 2
:; :
a
HH
σσ σσ
=≠ (claim)
b. N
d.f. 4=
D
d.f. 6=
06.23;F= Rejection region: F > 6.23
22. Population 1: Brand B
Population 2: Brand A
a. The claim is “the variance of the numbers of calories differ between the two brands.”
22 22
01 2 1 2
:; :
a
HH
σσ σσ
=≠ (claim)
b. N
d.f. 9=
D
d.f. 9=
CHAPTER 10 │ CHI-SQUARE TESTS AND THE
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23. Population 1: District 1
Population 2: District 2
a. The claim is “the standard deviations of science achievement test scores for eighth grade students
are the same in Districts 1 and 2.”
22 22
01 2 1 2
: (claim); :
a
HH
σσ σσ
=≠
2
d. Fail to reject 0.H
e. There is not enough evidence at the 10% level of significance to reject the administrator’s claim
that the standard deviations of science assessment test scores for eighth grade students are the
same in Districts 1 and 2.
24. Population 1: District 1
Population 2: District 2
D
05.20F= Rejection region: F > 5.20
c.
22
1
22
2
(33.9) 1.26
(30.2)
s
Fs
== ≈
d. Fail to reject 0.H
e. There is not enough evidence at the 1% level of significance to reject that the standard deviations
of U.S. history assessment test scores for eighth grade students are the same in Districts 1 and 2.
25. Population 1: New York
Population 2: California
a. The claim is “the standard deviation of the annual salaries for actuaries is greater in New York
than in California.”
430 CHAPTER 10 │ CHI-SQUARE TESTS AND THE
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26. Population 1: Florida
Population 2: Louisiana
D
01.985;F= Rejection region: F >1.985
c.
22
1
22
2
(10,100) 2.490
(6, 400)
s
Fs
== ≈
27. Hypothesis test results:
2
1:
σ
variance of population 1
2
2:
σ
variance of population 2
22
12
/
:
σσ
variance ratio
/
significance to reject the claim.
28. Hypothesis test results:
2
1:
σ
variance of population 1
2
2:
σ
variance of population 2
22
12
/
:
σσ
variance ratio
/
CHAPTER 10 │ CHI-SQUARE TESTS AND THE
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29. Hypothesis test results:
2
1:
σ
variance of population 1
2
2:
σ
variance of population 2
22
12
/
:
σσ
variance ratio
/
30. Hypothesis test results:
2
1:
σ
variance of population 1
2
2:
σ
variance of population 2
22
12
/
:
σσ
variance ratio
22
01 2
:/ 1H
σσ
=
22
A1 2
:/ 1H
σσ
>
Ratio
n1
n2
Sample
Ratio
F-Stat
P-value
22
12
/
σσ
7 13 1.1879483 1.1879483 0.3751
0.3751 0.01,P≈> so fail to reject 0.H There is not enough evidence at the 1% level of
significance to support the claim.
32. Right-tailed: R
F = 2.33
Left-tailed:
(1) N
d.f. = 15 and D
d.f. = 20
(2) F = 2.20
(3) Critical value is 11
0.45
2.20F=≈
.
432 CHAPTER 10 │ CHI-SQUARE TESTS AND THE
F
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34.
222 2 2
111 1 1
222 2 2
222 2 2
5.29 5.29
(0.331) (3.33) 0.485 4.880
3.61 3.61
LR
ss
FF
ss
σσσ
σσσ
⎛⎞ ⎛⎞
<< → << → <<
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
10.4 ANALYSIS OF VARIANCE
10.4 Try It Yourself Solutions
1a. The claim is “there is a difference in the mean a monthly sales among the sales regions.”
01234
:H
µ
µµµ
===
:
a
H At least one mean is different from the others. (claim)
b. 0.05
α
=
c. N
d.f. = 3
D
d.f. = 14
f. Reject 0.H
g. There is enough evidence at the 5% level of significance to conclude that there is a difference in
the mean monthly sales among the sales regions.
CHAPTER 10 │ CHI-SQUARE TESTS AND THE
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10.4 EXERCISE SOLUTIONS
1. 01 2 4
:H
µ
µµ
===…
:
a
H At least one mean is different from the others.
4. 0:
A
H There is no difference among the treatment means of Factor A.
:
aA
H There is at least one difference among the treatment means of Factor A.
0:
B
H There is no difference among the treatment means of Factor B.
:
aB
H There is at least one difference among the treatment means of Factor A.
0:
AB
H There is no interaction between Factor A and Factor B.
:
aAB
H There is interaction between Factor A and Factor B.
µ
c.
Variation
Sum of
Squares
Degrees of
Freedom
Mean
Squares
F
Between 0.518 2 0.259 1.02
Within 6.629 26 0.255
1.02F≈
d. Fail to reject 0.H
e. There is not enough evidence at the 5% level of significance to conclude that the mean costs per
ounce are different.
µ
434 CHAPTER 10 │ CHI-SQUARE TESTS AND THE
F
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c.
Variation
Sum of
Squares
Degrees of
Freedom
Mean
Squares
F
Between 198.529 2 99.265 0.418
Within 33.25 14 237.5
0.42F≈
d. Fail to reject 0.H
e. There is not enough evidence at the 5% level of significance to conclude that at least one of the
mean battery prices is different from the others.
c.
Variation
Sum of
Squares
Degrees of
Freedom
Mean
Squares
F
Between 3125.521 2 1562.761 21.99
Within 1918.441 27 71.053
21.99F≈
d. Reject 0.H
e. There is enough evidence at the 1% level of significance to conclude that at least one mean is
different.
0.27F≈
d. Fail to reject 0.H
e. There is not enough evidence at the 5% level of significance to conclude that at least one mean is
different.
CHAPTER 10 │ CHI-SQUARE TESTS AND THE
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d. Reject 0.H
e. There is enough evidence at the 1% level of significance to conclude that at least one mean cost
per miles is different.
10a. The claim is “at least one mean well-being index score is different.”
01234
:H
µ
µµµ
=== (claim)
:
a
H At least one mean is different from the others.
2.55F≈
d. Reject 0.H
e. There is enough evidence at the 10% level of significance to reject the claim that the mean
scores are the same for all regions.
11a. The claim is “the mean number of days patients spend at the hospital is the same for all four
regions.”
01234
:H
µ
µµµ
=== (claim)
:
a
H At least one mean is different from the others. (claim)
436 CHAPTER 10 │ CHI-SQUARE TESTS AND THE
F
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µ
c.
Variation
Sum of
Squares
Degrees of
Freedom
Mean
Squares
F
Between 274,390,504 5 54,878,100.8 1.63
Within 1,007,909,953 30 33,596,998.4
1.63F≈
d. Fail to reject 0.H
e. There is not enough evidence at the 5% level of significance to conclude that the mean salary is
different in at least one of the areas.
13a. The claim is “the mean energy consumption in at least one region is different from the others.”
01234
:H
µ
µµµ
===
:
a
H At least one mean is different from the others. (claim)
d. Reject 0.H
e. There is enough evidence at the 10% level of significance to conclude that the mean energy
consumption of at least one region is different from the others.
14a. The claim is “the mean amounts spent are equal for all regions.”
01234
:H
µ
µµµ
===
:
a
H At least one mean is different from the others. (claim)