SW388R6

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Chi-square Goodness of Fit Test
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Problem 1

• Based on the dataset GSS2000.SAV, is the
  following statement true, false, or an incorrect
  application of a statistic? Use 0.05 as the level
  of significance.
• The breakdown of survey respondents in the sample
  does not match the population breakdown in which
  51 were male and 49 were female. Specifically,
  survey respondents who were male were
  under-represented in the sample.
• 1. True
• 2. True with caution
• 3. False
• 4. Incorrect application of a statistic

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Request the chi-square goodness of fit test
To compute the chi-square goodness of fit test in
SPSS, select the Nonparametric Tests Chi-Square
command from the Analyze menu.
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Specify the variable to use in the analysis
First, move the variable "sex" to the Test
Variable List.
Second. click on the Values option button on the
Expected Values panel to tell SPSS that we will
enter the percentages to use for the expected
frequencies.
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Enter the percentage for the first category
First, type the percentage for the first
category, 51.
Second, click on the Add button to include this
percentage on the list.
The percentages must be entered in the order of
the data values for the categories. In our
problems, the percentages in the problem are
listed in the correct order.
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Enter the percentage for the second category
First, enter the percentage for the second
category, 49.
Second, click on the Add button to include this
percentage on the list.
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Complete the goodness of fit test request
With the variable and the breakdown for expected
frequencies entered, click on the OK button to
complete the request.
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Output for the goodness of fit test
This is the start of the output for the
chi-square goodness of fit test. Scroll down to
locate the information needed to answer our
question.
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Check the expected frequency assumption
The chi-square goodness of fit test assumes that
none of the expected frequencies are less than 5.
This assumption is evaluated by information in
the footnote of the test statistics table. For
this problem, we see than zero cells had an
expected frequency less than 5. The assumption is
satisfied.
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Check for statistical significance
To determine that the breakdown of the sample
differs from the specified population, the
probability of the test statistic must be less
than or equal to the level of significance. SPSS
labels this probability "Asymp. Sig." Since,
0.001 is less than 0.05, we reject the null
hypothesis and find support for the research
hypothesis that the sample frequency counts
differ from the expected frequency counts based
on the population.
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Check that the residual is interpreted correctly
A category in the sample is under-represented if
there are fewer than expected. This is indicated
by the residual (difference between the observed
frequency and actual frequency) being a negative
number. A category in the sample is
over-represented if there are more than
expected. This is indicated by the residual
being a positive number.
The residual for males is 26.7, indicating
under-representation. The answer to the question
is true.
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Problem 2

• Based on the dataset GSS2000.SAV, is the
  following statement true, false, or an incorrect
  application of a statistic? Use 0.05 as the level
  of significance.
• The breakdown of survey respondents in the sample
  does not match the population breakdown in which
  31 said they had a great deal of confidence in
  organized religion, 52 said they had only some
  confidence in organized religion and 17 said
  they had hardly any confidence in organized
  religion. Specifically, survey respondents who
  said they had a great deal of confidence in
  organized religion were under-represented in the
  sample.
• 1. True
• 2. True with caution
• 3. False
• 4. Incorrect application of a statistic

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Request the chi-square goodness of fit test
First, move the variable "conclerg" to the Test
Variable List.
Third, click on the OK button to complete the
request.
Second, enter the percentages for the expected
values. Note that we can enter decimal fractions
as well as percentage numbers.
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Output for the goodness of fit test
Since the "Asymp. Sig." 0f 0.124 is greater than
the level of significance of 0.05, we fail to
reject the null hypothesis and find that the
sample frequency counts do not differ from the
expected frequency counts based on the
population. The answer to the question is false.
The chi-square goodness of fit test assumes that
none of the expected frequencies are less than 5.
This assumption is evaluated by information in
the footnote of the test statistics table. For
this problem, we see than zero cells had an
expected frequency less than 5. The assumption is
satisfied.
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Problem 3

• Based on the dataset GSS2000.SAV, is the
  following statement true, false, or an incorrect
  application of a statistic? Use 0.05 as the level
  of significance.
• The breakdown of survey respondents in the sample
  does not match the population breakdown in which
  45 said that overall their marriages were very
  happy, 54 said that overall their marriages were
  pretty happy and 1 said that overall their
  marriages were not too happy. Specifically,
  survey respondents who said that overall their
  marriages were pretty happy were over-represented
  in the sample.
• 1. True
• 2. True with caution
• 3. False
• 4. Incorrect application of a statistic

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Request the chi-square goodness of fit test
First, move the variable "hapmar" to the Test
Variable List.
Third, click on the OK button to complete the
request.
Second, enter the percentages for the expected
values 45, 54, and 1.
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Output for the goodness of fit test
The chi-square goodness of fit test assumes that
none of the expected frequencies are less than 5.
This assumption is evaluated by information in
the footnote of the test statistics table. For
this problem, we see than 1 cell had an expected
frequency less than 5. The assumption is not
satisfied. The test should not be used.
The answer to the question is incorrect
application of a statistic.
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Steps in solving goodness of fit problems - 1
The following is a guide to the decision process
for answering chi-square goodness-of-fit
homework problems

• Are the minimum expected frequencies satisfied?
• No expected frequency less than 5

Incorrect application of a statistic
No
Yes
Is the probability of the chi-square test
statistic less than or equal to the level of
significance?
No
False
Yes
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Steps in solving goodness of fit problems - 2
Yes
Does the sign of the standardized residual
indicate that the category listed in the problem
was correctly cited as being over- or under-
represented.
No
False
Yes
True