2 Worksheet
Name: ________________________________

A
 2 goodness-of-fit compares the distribution of observed outcomes for a single categorical variable to the expected
outcomes predicted by a probability model to see if the model is viable. One variable, one sample. The null hypothesis is
that the probability model is correct. (All faces of the die are equally likely to occur.)

A
 2 test
of homogeneity compares observed distributions for several groups to each other to see if there is evidence of
difference among the respective populations. Several groups, but one variable. The null hypothesis is that the two groups
are drawn from populations in which the variable has the same model. (The distribution of eye color among boys is the same
as it is among girls.)

A
 2 test
of independence cross-categorizes one group on two variables to see if there is an association between them. One
sample, but two variables. The null hypothesis is that there is no association between the two variables. (Smoking is
independent of level of education.)
For # 1-3, state the type of test, the hypotheses, the test statistic, p-value, df, and decision.
1) Polly and Pete go through one page from a telephone book and write down the last digit of 50
numbers. Could the last digit from the phone numbers on this page be used as random digits?
Digit:
0
1
2
3
4
5
6
7
8
9
Freq:
7
6
3
2
5
7
2
10
8
0
X2 Goodness of fit test
x2 = 18
p-value = .0352
df = 9
telephone
Reject Ho
Ho: The proportion for each digit 0-9 is the same.
Ha: The proportion for at least one category is different.
2) The counseling unit of Woodrock College is interested in the relationship between anxiety level and the need to
succeed. A random sample of 200 college freshmen was taken. The freshmen were given tests to measure their anxiety
level and their need to succeed. Is there evidence that anxiety level and need to succeed are dependent?
Anxiety Level
Need
High
Medium
Low
High
30
40
5
Medium
17
50
33
Low
3
10
12
X2 Test for Independence
x2 = 29.6
p-value = .0000059
df = 4
Reject Ho
Ho: Anxiety level and need to succeed are independent for college freshmen.
Ha: Anxiety level and need to succeed are not independent.
3) In July 1991 and again in April 2001, the Gallup Poll asked random samples of 1015 adults about their opinions on
working parents. The table summarizes responses to the question “Considering the needs of both parents and children,
which of the following do you see as the ideal family in today’s society?”
Based on these results, do you think there was a
1991
2001
change in people’s attitudes during the 10 years
Both work full time
142
131
between these polls?
One works full time, other part time
274
244
One works, other works at home
One works, other stays home for kids
No opinion
152
396
51
173
416
51
X2 test for homogeneity
x2 = 4.03
p-value = .4019
to reject Ho
Ho: Adult opinions on working parents are the same for the two samples.
Ha: Adult opinions on working parents are different for the two samples.
df = 4 Fail
4) “Buddies” is a volunteer social service that works with disadvantaged children. Adults, 18 years & older, volunteer
from 1 to 6 hours per week to take disadvantaged children to the zoo, to a museum, to a movie, or to some other activity.
The program recruits adults from three main groups: college students, non-students living in the inner city, and nonstudents living in the suburbs. A random sample of adult volunteers gave the following information. Is there an
association between the number of hours volunteered and the type of volunteer?
Hours Volunteered
1-2
3-4
5-6
College student
115 (90.7)
93 (114.6)
47 (49.7)
Inner-city resident
88 (104.7)
150 (132.6)
56 (57.3)
Suburban resident
95 (102.5)
133 (129.4)
60 (56.1)
Assumptions:
Ho: The number of hours volunteered and the type of volunteer are independent.
Random sample stated
Ha: The number of hours volunteered and the type of volunteer are not independent.
All expected counts > 5 as shown in table
2  
(obs  exp)2
 16.7 with 4 df
exp
Chi-square test for independence
 = .05
p-value = x2cdf(16.7, ∞, 4) = .0022
Since p-value < α, we reject the null hypothesis. There is sufficient evidence to suggest that there
is an association between hours volunteered and type of volunteer.
5) In November 1997, the Gallup Organization released the results of a poll on family values. One of the questions asked
was: “For you personally, do you think it is necessary or not necessary to have a child at some point in your life
in order to feel fulfilled?” Results from adults in different countries appear below. Is there evidence that the
proportion of adults who would give each answer is different for each country?
U.S.
India
Mexico
Canada
Germany
Total
Yes
460 (616)
930 (616)
610 (616)
590 (616)
490 (616)
3080
No
510 (354)
60 (354)
380 (354)
370 (354)
450 (354)
1770
Undecided
30 (30)
10 (30)
10 (30)
40 (30)
60 (30)
1000
1000
Total
1000
1000
1000
150
5000
Assumptions:
Ho: The distribution of answers is the same for each country.
Random sample stated
Ha: The distribution of answers is different for at least one country.
All expected counts > 5 as shown in table
2  
(obs  exp)2
 628.08 with 8 df
exp
Chi-square test for homogeneity
 = .05
p-value = x2cdf(628.08, ∞, 8) = 0
Since p-value < α, we reject the null hypothesis. There is sufficient evidence to suggest that the
distribution of answers is not the same for at least one country.