Chapter 17: ChiSquare Tests
1. Nonparametric tests make few if any assumptions about the populations from which the data are obtained. For example, the populations do not need to form normal distributions, nor is it required that different populations in the same study have equal variances (homogeneity of variance assumption). Parametric tests require data measured on an interval or ratio scale. For nonparametric tests, any scale of measurement is acceptable.
2. a. The null hypothesis states that the gender distribution for theater goers is not different from
the distribution for the general population of students. For a sample of 600 students, the
expected frequencies are 330 females (55%) and 270 males (45%), and chisquare = 20.37.
With df = 1, the critical value is 3.84. Reject H_{0} and conclude that the gender distribution
for theater goers is significantly different from the distribution for the population of
students.
b. The null hypothesis states that the gender distribution for basketball fans is not different
from the distribution for the general population of students. For a sample of 180 students,
the expected frequencies are 99 females (55%) and 81 males (45%), and chisquare = 5.75.
With df = 1, the critical value is 3.84. Reject H_{0} and conclude that the gender distribution
for basketball fans is significantly different from the distribution for the population of
students.
3. a. The null hypothesis states that there is no preference among the four colors; p = 1/4 for all
categories. The expected frequencies are f_{e} = 15 for all categories, and chi square = 4.53.
With df = 3, the critical value is 7.81. Fail to reject H_{0} and conclude that there are no
significant preferences.
b. The results indicate that there are no significant preferences among the four colors, χ^{2}(3, N
= 60) = 4.53, p > .05.
4. a. The null hypothesis states that the age distribution for people who get speeding tickets is
not different from the distribution for the population of licensed drivers. With df = 1, the
critical value is 3.84. The expected frequencies are 48 over age 25 and 12 under age 25,
and chisquare = 10.42. Reject the null hypothesis and conclude that the age distribution
for people who receive speeding tickets is significantly different from the distribution for
the population of drivers.
b. The null hypothesis states that the age distribution for people who get parking tickets is
not different from the distribution for the population of licensed drivers. With df = 1, the
critical value is 3.84. The expected frequencies are 48 over age 25 and 12 under age 25,
and chisquare = 2.60. Fail to reject the null hypothesis and conclude that the age
distribution for people who receive speeding tickets is not significantly different from the
distribution for the population of drivers.
5. The null hypothesis states that wins and loses are equally likely. With 64 games, the expected frequencies are 32 wins and 32 losses. With df = 1 the critical value is 3.84, and the data produce a chisquare of 6.25. Reject the null hypothesis and conclude that home team wins are significantly more common that would be expected by chance.
6. The null hypothesis states that couples with the same initial do not occur more often than would be expected by chance. For a sample of 200, the expected frequencies are 13 with the same initial and 187 with different initials. With df = 1 the critical value is 3.84, and the data produce a chisquare of 2.96. Fail to reject the null hypothesis.
7. a. The null hypothesis states that couples with the same initial do not occur more often than
would be expected by chance. For a sample of 400, the expected frequencies are 26 with
the same initial and 374 with different initials. With df = 1 the critical value is 3.84, and
the data produce a chisquare of 5.92. Reject the null hypothesis.
b. A larger sample should be more representative of the population. If the sample continues
to be different from the hypothesis as the sample size increases, eventually the difference will be significant.
8. The null hypothesis states that the grade distribution for last semester has the same proportions as it did in 1985. For a sample of n = 200, the expected frequencies are 28, 52, 62, 38, and 20 for grades of A, B, C, D, and F, respectively. With df = 4, the critical value for chisquare is 9.49. For these data, the chisquare statistic is 6.68. Fail to reject H_{0} and conclude that there is no evidence that the distribution has changed.
9. a. H_{0} states that the distribution of automobile accidents is the same as the distribution of
registered drivers: 16% under age 20, 28% age 20 to 29, and 56% age 30 or older.
With df = 2, the critical value is 5.99. The expected frequencies for these three categories
are 48, 84, and 168. Chi square = 13.76. Reject H_{0} and conclude that the distribution of
automobile accidents is not identical to the distribution of registered drivers.
b. The chisquare test shows that the age distribution for people in automobile accidents is
significantly different from the age distribution of licensed drivers, χ^{2}(3, N= 180) =
13.76, p < .05.
10. a. The null hypothesis states that there is no advantage (no preference) for red or blue. With
df = 1, the critical value is 3.84. The expected frequency is 25 wins for each color, and
chisquare = 2.88. Fail to reject H_{0} and conclude that there is no significant advantage for
one color over the other.
b. The null hypothesis states that there is no advantage (no preference) for red or blue. With
df = 1, the critical value is 3.84. The expected frequency is 50 wins for each color, and
chisquare = 5.76. Reject H_{0} and conclude that there is a significant advantage for
the color red.
c. Although the proportions are identical for the two samples, the sample in part b is twice
as big as the sample in part a. The larger sample provides more convincing evidence of
an advantage for red than does the smaller sample.
11. The null hypothesis states that there are no preferences among the three designs; p = 1/3 for all categories. With df = 2, the critical value is 5.99. The expected frequencies are f_{e} = 40 for all categories, and chi square = 8.60. Reject H_{0} and conclude that there are significant preferences.
12. The null hypothesis states that the distribution of preferences is the same for both groups (same proportions). With df = 2, the critical value is 5.99. The expected frequencies are:
Design 1 Design 2 Design 3

Students

24

27

9

Older Adults

24

27

9

Chisquare = 7.94. Reject H_{0}.
13. The null hypothesis states that there is no relationship between the type of music and whether the women give their phone numbers. With df = 1, the critical value is 3.84. The expected frequencies are:
Phone Number No Number

Romantic Music

15

25

40

Neutral Music

15

25

40

30 50
Chisquare = 7.68. Reject H_{0}.
14. The null hypothesis states that the distribution of satisfaction scores is the same for both groups. With df = 1, the critical value is 3.84. The expected frequencies are:
Satisfied Not satisfied
Less reimbursement

55

45

100

Same or more reimbursement

33

27

60

88 72
Chisquare = 8.73. Reject H_{0}.
15. a. The null hypothesis states that the distribution of opinions is the same for those who live
in the city and those who live in the suburbs. For df = 1 and α = .05, the critical value for
chisquare is 3.84. The expected frequencies are:
Favor Oppose

For these data, chisquare = 3.12 Fail to reject H_{0 }and conclude that opinions in the city are not different from those in the suburbs.
b. The phi coefficient is 0.144.
16. a. The null hypothesis states that the distribution of opinions is the same for those who
live in the city and those who live in the suburbs. For df = 1 and α = .05, the critical
value for chisquare is 3.84. The expected frequencies are:
Favor Oppose

For these data, chisquare = 6.25. Reject H_{0 }and conclude that opinions in the city are different from those in the suburbs. The larger sample produces a significant relationship.
b. The phi coefficient is still 0.144. The sample size has no effect on the strength of the relationship.
17. a. The null hypothesis states that the proportion who falsely recall seeing broken glass
should be the same for all three groups. The expected frequency of saying yes is 9.67 for
all groups, and the expected frequency for saying no is 40.33 for all groups. With df = 2,
the critical value is 5.99. For these data, chisquare = 7.78. Reject the null hypothesis
and conclude that the likelihood of recalling broken glass is depends on the question that
the participants were asked.
b. Cramérs V = 0.228.
c. Participants who were asked abou the speed with the cars “smashed into” each other,
were more than two times more likely to falsely recall seeing broken glass.
d. The results of the chisquare test indicate that the phrasing of the question had a
significant effect on the participants’ recall of the accident, χ^{2}(2, N = 150) = 7.78, p < .05,
V = 0.228.
18. a. The null hypothesis states that the distribution of weights for men is the same as the
distribution for women. The expected frequencies are 81.6 desirable and 38.4
overweight for men, and 54.4 desirable and 25.6 overweight for women. With df = 1, the
critical value is 3.84. For these data, chisquare = 5.53. Reject the null hypothesis.
b. The phicoefficient is 0.166.
c. The chisquare test shows that the proportion of men who are overweight is significantly
greater than the proportion of women, χ^{2}(1, N = 200) = 5.53, p < .05, φ = 0.166.
19. The null hypothesis states that IQ and gender are independent. The distribution of IQ scores for boys should be the same as the distribution for girls. With df = 2 and and α = .05, the critical value is 5.99. The expected frequencies are 15 low IQ, 48 medium, and 17 high for both boys and girls. For these data, chisquare is 3.76. Fail to reject the null hypothesis. These data do not provide evidence for a significant relationship between IQ and gender.
20. The null hypothesis states that there is no relationship between dream content and gender; the distribution of aggression content should be the same for males and females. The critical value is 9.21. The expected frequencies are:
Low Medium High

Female

8.8

8.4

6.8

Male

13.2

12.6

10.2

The chisquare statistic is 25.52. Reject H_{0} with α = .01 and df = 2.
21. The null hypothesis states that there is no difference between the distribution of preferences predicted by women and the actual distribution for men. With df = 3 and = .05, the critical value is 7.81. The expected frequencies are:
somewhat slightly slightly somewhat
thin thin heavy heavy

Women

22.9

22.9

22.9

11.4

Men

17.1

17.1

17.1

8.6

Chi square = 9.13. Reject H_{0 }and conclude that there is a significant difference in the preferences predicted by women and the actual preferences expressed by men.
22. a. The null hypothesis states that there is no relationship between the personalities of the participants and the personalities of the avatars they create. With df = 1 and α = .05, the critical value is 3.84. The expected frequencies are:
Participant Personality
Introverted Extroverted

Introverted Avatar

17.1

27.9

45

Extroverted Avatar

20.9

34.1

55

38 62
The chisquare statistic is 4.12. Reject H_{0}.
b. The phicoefficient is 0.203.
23. a. The null hypothesis states that there is no relationship between IQ and volunteering. With
df = 2 and α = .05, the critical value is 5.99. The expected frequencies are:
IQ
High Medium Low

volunteer

37.5

75

37.5

not volunteer

12.5

25

12.5

The chisquare statistic is 4.75. Fail to reject H_{0} with α = .05 and df = 2.
24. a. The null hypothesis states that littering is independent of the amount of litter already on
the ground. With df = 2, the critical value is 5.99. The expected frequencies are:
Amount of Litter
None Small Large

Litter

31.33

31.33

31.33

Not Litter

58.67

58.67

58.67

Chisquare = 25.88. Reject H_{0}.
b. V = 0.310 (a medium effect)
25. The null hypothesis states that there is no relationship between the season of birth and schizophrenia. With df = 3 and = .05, the critical value is 7.81. The expected frequencies are:
Summer Fall Winter Spring

No Disorder

23.33

23.33

26.67

26.67

Schizophrenia

11.67

11.67

13.33

13.33

Chi square = 3.62. Fail to reject H_{0 }and conclude that these data do not provide enough evidence to conclude that there is a significant relationship between the season of birth and schizophrenia.
Solutions  Chapter 17  page
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