CHAPTER
Chi-Square Tests and the
F
-Distribution
401
1
0
10.1 GOODNESS OF FIT TEST
10.1 Try It Yourself Solutions
1.
Tax Preparation Method % of people Expected frequency
Accountant
25%
500(0.25) = 125
2a. The expected frequencies are 64, 80, 32, 56, 60, 48, 40, and 20, all of which are at least 5.
b. Claimed distribution:
Ages Distribution
0−9 16%
10−19 20%
0:HThe distribution of ages is as shown in table above.
:
a
H The distribution of ages differs from the claimed distribution. (claim)
c. 0.05
α
= d. d.f. = n − 1 = 7
e. 2
0
χ
= 14.067; ion region: 214.067
χ
>
f.
Ages Distribution Observed Expected OE
E
2
()−
0−9 16% 76 64 2.250
10−19 20% 84 80 0.200
26.694
χ
≈
g. Fail to reject 0.H
h. There is not enough evidence at the 5% level of significance to support the sociologist’s claim
that the distribution of ages differs from the age distribution 10 years ago.
402 CHAPTER 10 │ CHI-SQUARE TESTS AND THE
F
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3a. The expected frequency for each category is 30 which is at least 5.
b. Claimed distribution:
Color Distribution
Brown 16.6%
Yellow 16.6%
c. 0.05
α
=
d. d.f. = n − 1 = 5
e. 2
0
χ
= 11.071; Rejection region: 211.071
χ
>
f.
Color Distribution Observed Expected OE
E
2
()−
Brown 16.6% 22 30 2.133
Yellow 16.6% 27 30 0.300
212.933
χ
≈
CHAPTER 10 │ CHI-SQUARE TESTS AND THE
F
-DISTRIBUTION 403
10.1 EXERCISE SOLUTIONS
3. 11
(150)(0.3) 45Enp== = 4. 11
(500)(0.9) 450Enp== =
5. 11
(230)(0.25) 57.5Enp== = 6. 11
(415)(0.08) 33.2Enp== =
7a. Claimed Distribution:
Age Distribution
2−17 26.7%
18−24 19.8%
25−39 19.7%
40−49 14%
50+ 19.8%
c.
Age Distribution Observed Expected OE
E
2
()−
2−17 26.7% 240 267 2.730
18−24 19.8% 214 198 1.293
404 CHAPTER 10 │ CHI-SQUARE TESTS AND THE
F
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8a. Claimed distribution:
Response Distribution
2 cups a week 15%
1 cup a week 13%
b. 2
0
χ
= 7.815, Rejection region: 27.815
χ
>
c.
Response Distribution Observed Expected OE
E
2
()−
2 cups a week 15% 206 240 4.817
8.483
28.483
χ
≈
d. Reject 0.H
e. There is enough evidence at the 5% level of significance to conclude that the distribution of the
amounts of coffee people drink differs from the claimed distribution.
9a. Claimed distribution:
Day Distribution
Sunday 7%
Monday 4%
Tuesday 6%
χ
CHAPTER 10 │ CHI-SQUARE TESTS AND THE
F
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c.
Distribution Observed Expected OE
E
2
()−
Sunday 7% 43 35 1.829
Monday 4% 16 20 0.800
217.595
χ
≈
d. Reject 0.H
e. There is enough evidence at the 1% level of significance to conclude that the distribution of
delivery days has changed.
10a.Claimed distribution:
Response Distribution
Limited advancement 41%
a
b. 2
0
χ
=13.277, Rejection region: 213.277
χ
>
c.
Response Distribution Observed Expected OE
E
2
()−
Limited advancement 41% 78 82.00 0.195
406 CHAPTER 10 │ CHI-SQUARE TESTS AND THE
F
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11a.Claimed distribution:
Season Distribution
Spring 25%
0
c.
Season Distribution Observed Expected OE
E
2
()−
Spring 25% 312 300 0.480
d. Fail to reject 0.H
e. There is not enough evidence at the 5% level of significance to reject the claim that the
distribution of the number of homicide crimes in California by season is uniform.
12a. Claimed distribution:
Month Distribution Month Distribution
January 8.33% July 8.33%
February 8.33% August 8.33%
CHAPTER 10 │ CHI-SQUARE TESTS AND THE
F
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c.
Month Distribution Observed Expected OE
E
2
()−
January 8.33% 98 100 0.040
February 8.33% 103 100 0.090
August 8.33% 109 100 0.810
September 8.33% 112 100 1.440
October 8.33% 95 100 0.250
13a. Claimed distribution:
Month Distribution
Strongly agree 55%
Somewhat agree 30%
Neither agree or disagree 5%
b. 2
0
χ
= 9.488, Rejection region: 29.488
χ
>
c.
Month Distribution Observed Expected OE
E
2
()−
Strongly agree 55% 86 110 5.236
Somewhat agree 30% 62 60 0.067
408 CHAPTER 10 │ CHI-SQUARE TESTS AND THE
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d. Reject 0.H
e. There is enough evidence at the 5% level of significance to conclude that the distribution of
opinions of U.S. parents on whether a college education is worth the expense differs from the
claimed or expected distribution.
14a. Claimed distribution:
b. 2
0
χ
= 4.605, Rejection region: 24.605
χ
>
c.
Response Distribution Observed Expected
OE
E
2
()−
Saving for your child’s college education 50% 180 200 2.000
Saving for your own retirement 37% 172 148 3.892
Not sure 13% 48 52 0.308
6.200
26.200
χ
≈
15a. Claimed distribution:
Response Distribution
Larger 33.3%
Same size 33.3%
Smaller 33.3%
χ
CHAPTER 10 │ CHI-SQUARE TESTS AND THE
F
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c.
Response Distribution Observed Expected OE
E
2
()−
210.308
χ
≈
d. Reject 0.H
e. There is enough evidence at the 5% level of significance to reject the claim that the distribution of
prospective home buyers by the size they want their next house to be is uniform.
16a. Claimed distribution:
Day Distribution
Sunday 2
14 %
7
Wednesday 2
14 %
7
Thursday 2
14 %
7
Friday 2
14 %
410 CHAPTER 10 │ CHI-SQUARE TESTS AND THE
F
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c.
Day Distribution Observed Expected OE
E
2
()−
Sunday 2
14 %
7 65 100 12.250
Monday 2
14 %
7 103 100 0.090
26.800
226.800
χ
≈
d. Reject 0.H
e. There is enough evidence at the 1% level of significance to reject the claim that the distribution of
the number of births by day of the week is uniform.
18. Chi-Square goodness-of-fit results:
Observed: Not married
Expected: Married
N DF Chi-Square P-Value
250 7 9.523134 0.2172
CHAPTER 10 │ CHI-SQUARE TESTS AND THE
F
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19a.Frequency distribution: 69.435; 8.337
µ
σ
=≈
0:H Test scores have a normal distribution. (claim)
:
a
H Test scores do not have a normal distribution.
b. 2
0
χ
= 13.277; Rejection region . 213.277
χ
>
c. 2
χ
= 0.613
20a.Frequency distribution: 74.775; 9.822
µ
σ
=≈
Lower
Boundary
Upper
Boundary
Lower
z-score
Upper
z-score
Area
50.5 60.5
−2.47 −1.45 0.0668
60.5 70.5
−1.45 −0.44 0.2564
70.5 80.5
−0.44 0.58 0.3891
80.5 90.5 0.58 1.60 0.2262
90.5 100.5 1.60 2.62 0.0504
Lower
Boundary
Upper
Boundary
Lower
z-score
Upper
z-score
Area
49.5 58.5
Class
Boundaries
Distribution
Frequency
Expected
OE
E
2
()
−
49.5−58.5 8.67% 19 17 0.2353
58.5−67.5 31.39% 61 63 0.0635
412 CHAPTER 10 │ CHI-SQUARE TESTS AND THE
F
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0:H Test scores have a normal distribution. (claim)
:
a
H Test scores do not have a normal distribution.
10.2 INDEPENDENCE
10.2 Try It Yourself Solutions
1a.
Hotel Leg Room Rental Size Other Total
Business 36 108 14 22 180
Leisure 38 54 14 14 120
Total 74 162 28 36 300
b. n = 300
2a. The claim is that “the travel concerns depend on the purpose of travel.”
0:H Travel concern is independent of travel purpose.
:
a
H Travel concern is dependent on travel purpose. (claim)
b. 0.01
α
= c. (r − 1)(c − 1) = 3
CHAPTER 10 │ CHI-SQUARE TESTS AND THE
F
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f. Fail to reject 0.H
g. There is not enough evidence to conclude that travel concern is dependent on travel purpose.
c. 2
09.210;
χ
= Rejection region: 29.210
χ
>
d. 215.306
χ
≈
10.2 EXERCISE SOLUTIONS
1. Find the sum of the row and the sum of the column in which the cell is located. Find the product
of these sums. Divide the product by the sample size.
2. In a contingency table, a marginal frequency is the frequency an entire category of a variable
occurs, whereas a joint frequency is a frequency from a cell in the interior of a contingency table.
3. Answer will vary. Sample answer: For both the chi-square test for independence and the chi-
square goodness-of-fit test, you are testing a claim about data that are in categories. However, the
4. A chi-square independence test is always a right-tailed test because if the variables are dependent,
then the chi-square test statistic will be large, which is evidence for rejecting the null hypothesis.
414 CHAPTER 10 │ CHI-SQUARE TESTS AND THE
F
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6. True.
7a.
Athlete has
Result Stretched Not stretched Total
Inquiry 18 22 40
No Inquiry 211 189 400
229 211 440
8a.
Athlete has
Result Drug Placebo Total
Nausea 36 13 49
No Nausea 254 262 516
Total 290 275 565
9a.
Preference
Bank employee
New
procedure
Old
Procedure
No preference
Total
Teller 92 351 50 493
Customer service 76 42 8 126
Total 168 393 58 619
b.
Preference
Bank employee New
procedure
Old
Procedure
No preference
Teller 133.80 313.00 46.19
Customer service 34.20 80.00 11.81
CHAPTER 10 │ CHI-SQUARE TESTS AND THE
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b.
Rating
Size Excellent Fair Poor
Seats 100 or fewer 165.92 235.58 148.50
Seats over 100 196.08 278.42 175.50
12a.
Age
Type of
movie rented
18−24
25−34
35−44
45−64
65 and older
Total
Comedy 38 30 24 10 8 110
Action 15 17 16 9 5 62
Drama 12 11 19 25 13 80
Total 65 58 59 44 26 252
416 CHAPTER 10 │ CHI-SQUARE TESTS AND THE
F
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c.
O E O
−
E OE
2
()− OE
E
2
()
−
43 41.129 1.871 3.500641 0.0851
42 41.905 0.095 0.009025 0.0002
20.297
χ
≈
d. Fail to reject 0.H There is not enough evidence at the 1% level of significance to reject the
claim that skill level in a subject is independent of location.
14a. The claim is “attitudes about the safety steps taken by the school staff are related to the type of
school.”
c.
O E O
−
E OE
2
()− OE
E
2
()
−
41 50.074
−10.074 101.4855 2.0267
51 40.926 10.074 101.4855 2.4797
64 53.926 10.074 101.4855 1.8819
34 44.074
−10.074 101.4855 2.3026
8.6909
28.691
χ
≈
d. Reject 0.H There is enough evidence at the 1% level of significance to conclude that attitudes
about safety are dependent on the type of school.
15a. The claim is “the number of times former smokers tried to quit before they were habit-free is
related to gender.”
CHAPTER 10 │ CHI-SQUARE TESTS AND THE
F
-DISTRIBUTION 417
c.
O E O
−
E OE
2
()− OE
E
2
()
−
271 270.930 0.070 0.004900 0
0.0018
20.002
χ
≈
d. Fail to reject 0.H There is not enough evidence at the 5% level of significance to conclude that
the number of times former smokers tried to quit is dependent on gender.
0
c.
O E O
−
E OE
2
()− OE
E
2
()
−
97 99.0 −2.0 4.00 0.0404
42 37.5 4.5 20.25 0.5400
26 25.5 0.5 0.25 0.0098
3.4304
23.430
χ
≈
d. Fail to reject 0.H There is not enough evidence at the 5% level of significance to conclude that
the adults’ rating of the movie are dependent on gender.
418 CHAPTER 10 │ CHI-SQUARE TESTS AND THE
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c.
O E O
−
E OE
2
()− OE
E
2
()
−
39 31.660 7.34 53.8756 1.7017
25 32.340 −7.34 53.8756 1.6659
54 61.340 −7.34 53.8756 0.8783
70 62.660 7.34 53.8756 0.8598
5.1057
25.106
χ
≈
d. Reject 0.H There is enough evidence at the 10% level of significance to conclude that results
are dependent on the type of treatment.
Sample answer: Because the results are significant at the 10% level of significance, you may
consider recommending the drug. However, you may want to perform more tests before
recommending the drug.
0
c.
O E O
−
E OE
2
()− OE
E
2
()
−
58 66.667 −8.667 75.116889 1.1267
81 66.667 14.333 205.434889 3.0815
61 66.667 −5.667 32.114889 0.4817
42 33.333 8.667 75.116889 2.2535
19 33.333 −14.333 205.434889 6.1631
39 33.333 5.667 32.114889 0.9635
14.0700
214.070
χ
≈
d. Reject 0.H There is enough evidence at the 10% level of significance to conclude that the
result is dependent on the type of treatment.
CHAPTER 10 │ CHI-SQUARE TESTS AND THE
F
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c.
O E O
−
E OE
2
()− OE
E
2
()
−
30 39.421 −9.421 88.755241 2.2515
36 31.230 4.770 22.752900 0.7286
7.3256
27.326
χ
≈
d. Fail to reject 0.H There is not enough evidence at the 1% level of significance to conclude that
the reason(s) for continuing education are dependent on the type or worker. Based on these
results, marketing strategies should not differ between technical and non-technical audiences in
regard to reason(s) for continuing education.
c.
O E O
−
E OE
2
()− OE
E
2
()
−
31 27.6602 3.3398 11.154264 0.4033
22 24.0680 −2.0680 4.276624 0.1777
21 22.2718 −1.2718 1.617475 0.0726
5.7572
25.757
χ
≈
d. Fail to reject 0.H There is not enough evidence at the 1% level of significance to conclude that
age is related to which aspect of career development is considered to be most important.
420 CHAPTER 10 │ CHI-SQUARE TESTS AND THE
F
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b. d.f. = (r − 1)(c − 1) = 2
2
χ
= 5.991; Rejection region: 2
χ
> 5.991
2144.099
χ
≈
d. Reject 0.H There is enough evidence at the 5% level of significance to conclude that the type
of crash is dependent on the type of vehicle.
22a. The claim is “the metropolitan status of libraries and Internet access speed are related.”
0:H Metropolitan status of libraries and Internet access speed are independent..
χ
χ
c.
O E O
−
E OE
2
()− OE
E
2
()
−
5 14.4921 −9.4921 90.099962 6.2172
20 27.4471 −7.4471 55.459298 2.0206
58 41.0608 16.9392 286.936497 6.9881
24 23.5714 0.4826 0.183698 0.0078