SW388R6

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Chi-square Test of Independence
2
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. Base your answer on the output
  for a chi-square test of independence with an
  analysis of standardized residuals.
• For the population represented by this sample of
  survey respondents, there is a relationship
  between opinion about the legalization of
  marijuana and strength of religious affiliation.
  Specifically, among survey respondents who
  thought the use of marijuana should be made
  legal, there were more who said they did not have
  a religious affiliation than would be expected.
• 1. True
• 2. True with caution
• 3. False
• 4. Incorrect application of a statistic

3
Request the chi-square test of independence
To compute the chi-square test of independence in
SPSS, select the Descriptive Statistics
Crosstabs command from the Analyze menu.
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Specify the variables to use in the analysis
First, move the dependent variable reliten" to
the Row(s) variable list.
Second, move the independent variable grass to
the Column(s) variable list.
Third, click on the Statistics button to select
the chi-square test.
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Select the chi-square statistic
First, mark the checkbox for the Chi-square
statistic.
Second, click on continue button to close the
dialog.
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Specify the contents of the cells
First, click on the Cells button to access the
dialog box to specify the cell contents.
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Include expected counts and residuals in output
First, mark the checkbox for expected counts, in
addition to the default of observed counts.
Third, click on Continue button to close the
dialog.
Second, mark the checkbox for standardized
residuals that we will use to determine
significance of individual cell values.
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Complete the test of independence request
Click on the OK button to complete the request.
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Output for the test of independence
The research hypothesis for the chi-square test
of independence states that differences in
opinion about the legalization of marijuana are
related to differences in strength of religious
affiliation, i.e., the actual frequencies do not
equal the expected frequencies for one or more
cells in the cross-tabulated table. The null
hypothesis states that differences in opinion
about the legalization of marijuana are
independent of differences in strength of
religious affiliation, i.e., the actual
frequencies equal the expected frequencies in the
cross-tabulated table.
The Chi-square test of Independence can be used
with variables at all levels of measurement.
10
Check the expected frequency assumption
The chi-square test of independence 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 that zero cells had an
expected frequency less than 5. The assumption is
satisfied.
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Check for statistical significance
The probability of the test statistic is 0.029.
Since this probability is less than or equal to
the level of significance of 0.05, we reject the
null hypothesis and conclude that the analysis
supports the research hypothesis that differences
in opinion about the legalization of marijuana
are related to differences in strength of
religious affiliation.
12
Significance of the standardized residual
The standardized residual for the cell in the
column labeled "LEGAL," in the row labeled "NO
RELIGION" is 2.1. This is greater than the
critical value of 1.96 associated with the level
of significance of 0.05. The specific
relationship between survey respondents who
thought the use of marijuana should be made legal
and survey respondents who said they did not have
a religious affiliation is supported.
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The sign of the standardized residual
The sign of the standardized residual (2.1) is
positive indicating that there were more survey
respondents who said they did not have a
religious affiliation than would be expected
among survey respondents who thought the use of
marijuana should be made legal. 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. Base your answer on the output
  for a chi-square test of independence with an
  analysis of standardized residuals.
• For the population represented by this sample of
  survey respondents, there is a relationship
  between sex and computer use. Specifically, among
  survey respondents who were male, there were more
  who said they didn't use a computer than would be
  expected.
• 1. True
• 2. True with caution
• 3. False
• 4. Incorrect application of a statistic

15
Output for the test of independence
The research hypothesis for the chi-square test
of independence states that differences in sex
are related to differences in computer use, i.e.,
the actual frequencies do not equal the expected
frequencies for one or more cells in the
cross-tabulated table. The null hypothesis
states that differences in sex are independent of
differences in computer use, i.e., the actual
frequencies equal the expected frequencies in the
cross-tabulated table.
The Chi-square test of Independence can be used
with variables at all levels of measurement.
16
Check the expected frequency assumption
The chi-square test of independence 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 that zero cells had an
expected frequency less than 5. The assumption is
satisfied.
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Check for statistical significance
The probability of the test statistic is 0.511.
Since this probability is greater than the level
of significance of 0.05, we fail to reject the
null hypothesis and conclude that the analysis
does not support the research hypothesis that
differences in sex are related to differences in
computer use. The answer to the question is
false.
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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. Base your answer on the output
  for a chi-square test of independence with an
  analysis of standardized residuals.
• For the population represented by this sample of
  survey respondents, there is a relationship
  between sex and happiness of marriage.
  Specifically, among survey respondents who were
  female, there were fewer who said that overall
  their marriages were very happy than would be
  expected.
• 1. True
• 2. True with caution
• 3. False
• 4. Incorrect application of a statistic

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Output for the test of independence
The research hypothesis for the chi-square test
of independence states that differences in
opinion about the legalization of marijuana are
related to differences in strength of religious
affiliation, i.e., the actual frequencies do not
equal the expected frequencies for one or more
cells in the cross-tabulated table. The null
hypothesis states that differences in opinion
about the legalization of marijuana are
independent of differences in strength of
religious affiliation, i.e., the actual
frequencies equal the expected frequencies in the
cross-tabulated table.
The Chi-square test of Independence can be used
with variables at all levels of measurement.
20
Check the expected frequency assumption
The Chi-square test of Independence assumes that
the expected frequencies for all cells in the
frequency distribution are 5 or higher. The
footnote to the table of test statistics states
that 2 cells had an expected frequency less than
5. This requirement is not satisfied. The answer
to the question is incorrect application of a
statistic
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Problem 4

• 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. Base your answer on the output
  for a chi-square test of independence with an
  analysis of standardized residuals.
• For the population represented by this sample of
  survey respondents, there is a relationship
  between sex and frequency of reading the
  newspaper. Specifically, among survey respondents
  who were male, there were more who said they read
  the newspaper a few times a week than would be
  expected.
• 1. True
• 2. True with caution
• 3. False
• 4. Incorrect application of a statistic

22
Output for the test of independence
The research hypothesis for the chi-square test
of independence states that differences in sex
are related to differences in frequency of
reading the newspaper, i.e., the actual
frequencies do not equal the expected frequencies
for one or more cells in the cross-tabulated
table. The null hypothesis states that
differences in sex are independent of differences
in frequency of reading the newspaper, i.e., the
actual frequencies equal the expected frequencies
in the cross-tabulated table.
The Chi-square test of Independence can be used
with variables at all levels of measurement.
23
Check the expected frequency assumption
The chi-square test of independence 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 that
zero cells had an expected frequency less than 5.
The assumption is satisfied.
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Check for statistical significance
The probability of the test statistic is 0.026.
Since this probability is less than or equal to
the level of significance of 0.05, we reject the
null hypothesis and conclude that the analysis
supports the research hypothesis that differences
in sex are related to differences in frequency of
reading the newspaper.
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Significance of the standardized residual
The standardized residual for the cell in the
column labeled "MALE," in the row labeled "FEW
TIMES A WEEK" is 1.8. This is less than the
critical value of 1.96 associated with the level
of significance of 0.05. The specific
relationship between survey respondents who were
male and survey respondents who said they read
the newspaper a few times a week is not
supported. The answer to the question is false.
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Solving test of independence problems - 1
The following is a guide to the decision process
for answering chi-square test of independence
homework problems

• Is the minimum expected frequency requirement
  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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Solving test of independence problems - 2

• Is the standardized residual larger than the
  critical value for the level of significance?
• Level of significance 0.05 1.96
• Level of significance 0.01 2.58

No
False
Yes
Does the sign of the standardized residual
indicate that the observed count was more or less
than expected, as stated in the problem?
No
False
Yes
True