Instructor Manual For
Essentials of Statistics for the
Behavioral Sciences, 10th Edition By
Frederick Gravetter, Larry Wallnau,
Lori-Ann Forzano, James Witnauer
(All Chapters 1-15, 100% Original
Verified, A+ Grade)
All Chapters Arranged Reverse: 15-1
This is The Original Instructor
Manual For 10th Edition, All other
Files in The Market are
Fake/Old/Wrong Edition.
, Chapter 15: The Chi-Square Statistic: Tests for Goodness of Fit
and Independence
Chapter Outline
15.1 Introduction to Chi-Square: The Test for Goodness of Fit
Parametric and Nonparametric Statistical Tests
The Chi-Square Test for Goodness of Fit
The Null Hypothesis for the Goodness-of-Fit Test
The Data for the Goodness-of-Fit Test
Expected Frequencies
The Chi-Square Statistic
15.2 An Example of the Chi-Square Test for Goodness of Fit
The Chi-Square Distribution and Degrees of Freedom
Locating the Critical Region for a Chi-Square Test
A Complete Chi-Square Test for Goodness of Fit
In the Literature: Reporting the Results for Chi-Square
Goodness of Fit and the Single-Sample t Test
15.3 The Chi-Square Test for Independence
The Null Hypothesis for the Test for Independence
Observed and Expected Frequencies
The Chi-Square Statistic and Degrees of Freedom
A Summary of the Chi-Square Test for Independence
15.4 Effect Size and Assumptions for the Chi-Square Tests
Cohen’s w
The Phi-Coefficient and Cramér’s V
Assumptions and Restrictions for Chi-Square Tests
Learning Objectives and Chapter Summary
1. Describe parametric and nonparametric hypothesis tests.
2. Describe the data (observed frequencies) for a chi-square test for goodness of fit.
3. Describe the hypotheses for a chi-square test for goodness of fit, explain how the
expected frequencies are obtained, and find the expected frequencies for a specific
research example.
4. Define the degrees of freedom for the chi-square test for goodness of fit and locate the
critical value for a specific alpha level in the chi-square distribution.
5. Conduct a chi-square test for goodness of fit and report the results as they would appear
in the scientific literature.
6. Define the degrees of freedom for the chi-square test for independence and locate the
critical value for a specific alpha level in the chi-square distribution.
7. Describe a chi-square test for independence and explain how the expected frequencies are
obtained.
8. Conduct a chi-square test for independence and report the results as they would appear in
the scientific literature.
9
, 9. Compute Cohen’s w to measure effect size for both chi-square tests.
10. Compare the phi-coefficient or Cramér’s V to measure effect size for the chi-square test
for independence.
11. Identify the basic assumptions and restrictions for chi-square tests.
The following synthesizes the key ideas and takeaways from this chapter:
1. Students should recognize the research situations in which a chi-square test is appropriate.
Chi-square tests are intended for research questions concerning the proportion of the
population in different categories. For a chi-square test, there is not a numerical score for
each individual, and you do not compute a sample mean or a sample variance. Instead,
each individual is classified into a category, and the number of individuals in each
category is counted. The resulting data are called observed frequencies.
2. Students should be able to conduct a chi-square test for goodness of fit to evaluate a
hypothesis about the shape (proportions) of a population distribution.
The data for the chi-square test for goodness of fit consist of a sample of individuals who
have been classified into categories of one variable. The numbers of individuals in each
category are called the observed frequencies. The null hypothesis for the chi-square test
for goodness of fit typically falls into one of two types: (1) a no-preference hypothesis
that states that the population is distributed evenly across the categories, or (2) a no-
difference hypothesis that states that the population distribution is not different from an
established distribution. In either case, the proportions from the null hypothesis are used
to construct an ideal sample distribution, called expected frequencies, and then the chi-
square statistic is computed to determine how well the data (observed frequencies) fit the
hypothesis (expected frequencies). The greater the discrepancy between the data and the
hypothesis, the greater the value for chi-square and the likelihood of rejecting the null
hypothesis.
3. Students should be able to conduct a chi-square test for independence to evaluate the
relationship between two variables in the population.
For the chi-square test for independence, each individual in the sample is classified into
one category for each of two different variables. The categories of one variable form the
rows of a data matrix and the categories of the second variable form the columns. The
number of individuals in each cell of the matrix is the observed frequency for that cell.
The null hypothesis for the chi-square test for independence can be phrased two ways:
(1) there is no relationship between the two variables (they are independent), or (2) the
distribution for one variable has the same proportions for all the categories of the second
variable. Expected frequencies representing an ideal sample distribution are computed
from the null hypothesis, and the chi-square statistic is computed to determine how well
the data (observed frequencies) fit the hypothesis (expected frequencies).
4. Students should be able to evaluate the effect size (strength of relationship) for a chi-square
test of independence by computing Cohen’s w, a phi-coefficient (for a 2x2 data matrix) or
10
, Cramér’s V (for a larger data matrix).
The measures of effect size are interpreted the same as correlations and provide a
measure of the strength of the relationship.
Other Lecture Suggestions
1. For some reason, students often assume that the null hypothesis for the chi-square test for
goodness of fit always specifies equal proportions. It helps to remind them that the no-difference
hypothesis exists and that it usually produces unequal proportions across the categories. A good
classroom example compares the distribution of licensed drivers across age categories with the
distribution of automobile accidents across age categories (see problem 4 at the end of the
chapter).
2. Although the no-preference hypothesis applies to simple preference situations (like a taste test
among four brands of cola), it also applies to other areas such as questions of perceptual
discrimination. For example, people could be shown different modifications of a photograph
(eyes moved apart, mouth widened, etc.) and asked to select the most attractive. Or, infants could
be shown visual patters with different levels of complexity to determine whether they prefer to
look at one over the others.
3. When computing expected frequencies for the chi-square test for independence, point out that
the values at the bottom of each column are not free to vary but are restricted by the column
totals. Similarly, the final values in each row are restricted. If you remove the restricted values,
you are left with a matrix with (R-1) rows and (C-1) columns, hence df = (R-1)(C-1). Also note
that if students are computing the expected frequencies individually, they only need to compute a
number of values equal to the degrees of freedom: the values at the bottom of each column and at
the end of each row can be found by subtraction from the row and column totals.
4. Remind students that the expected frequencies for the chi-square test for independence should
have the same proportions across each row, not necessarily the same frequencies. To check that
they have the correct fe values, they should compute one or two proportions within a row and
compare them with the corresponding proportions computed within a different row.
11
Essentials of Statistics for the
Behavioral Sciences, 10th Edition By
Frederick Gravetter, Larry Wallnau,
Lori-Ann Forzano, James Witnauer
(All Chapters 1-15, 100% Original
Verified, A+ Grade)
All Chapters Arranged Reverse: 15-1
This is The Original Instructor
Manual For 10th Edition, All other
Files in The Market are
Fake/Old/Wrong Edition.
, Chapter 15: The Chi-Square Statistic: Tests for Goodness of Fit
and Independence
Chapter Outline
15.1 Introduction to Chi-Square: The Test for Goodness of Fit
Parametric and Nonparametric Statistical Tests
The Chi-Square Test for Goodness of Fit
The Null Hypothesis for the Goodness-of-Fit Test
The Data for the Goodness-of-Fit Test
Expected Frequencies
The Chi-Square Statistic
15.2 An Example of the Chi-Square Test for Goodness of Fit
The Chi-Square Distribution and Degrees of Freedom
Locating the Critical Region for a Chi-Square Test
A Complete Chi-Square Test for Goodness of Fit
In the Literature: Reporting the Results for Chi-Square
Goodness of Fit and the Single-Sample t Test
15.3 The Chi-Square Test for Independence
The Null Hypothesis for the Test for Independence
Observed and Expected Frequencies
The Chi-Square Statistic and Degrees of Freedom
A Summary of the Chi-Square Test for Independence
15.4 Effect Size and Assumptions for the Chi-Square Tests
Cohen’s w
The Phi-Coefficient and Cramér’s V
Assumptions and Restrictions for Chi-Square Tests
Learning Objectives and Chapter Summary
1. Describe parametric and nonparametric hypothesis tests.
2. Describe the data (observed frequencies) for a chi-square test for goodness of fit.
3. Describe the hypotheses for a chi-square test for goodness of fit, explain how the
expected frequencies are obtained, and find the expected frequencies for a specific
research example.
4. Define the degrees of freedom for the chi-square test for goodness of fit and locate the
critical value for a specific alpha level in the chi-square distribution.
5. Conduct a chi-square test for goodness of fit and report the results as they would appear
in the scientific literature.
6. Define the degrees of freedom for the chi-square test for independence and locate the
critical value for a specific alpha level in the chi-square distribution.
7. Describe a chi-square test for independence and explain how the expected frequencies are
obtained.
8. Conduct a chi-square test for independence and report the results as they would appear in
the scientific literature.
9
, 9. Compute Cohen’s w to measure effect size for both chi-square tests.
10. Compare the phi-coefficient or Cramér’s V to measure effect size for the chi-square test
for independence.
11. Identify the basic assumptions and restrictions for chi-square tests.
The following synthesizes the key ideas and takeaways from this chapter:
1. Students should recognize the research situations in which a chi-square test is appropriate.
Chi-square tests are intended for research questions concerning the proportion of the
population in different categories. For a chi-square test, there is not a numerical score for
each individual, and you do not compute a sample mean or a sample variance. Instead,
each individual is classified into a category, and the number of individuals in each
category is counted. The resulting data are called observed frequencies.
2. Students should be able to conduct a chi-square test for goodness of fit to evaluate a
hypothesis about the shape (proportions) of a population distribution.
The data for the chi-square test for goodness of fit consist of a sample of individuals who
have been classified into categories of one variable. The numbers of individuals in each
category are called the observed frequencies. The null hypothesis for the chi-square test
for goodness of fit typically falls into one of two types: (1) a no-preference hypothesis
that states that the population is distributed evenly across the categories, or (2) a no-
difference hypothesis that states that the population distribution is not different from an
established distribution. In either case, the proportions from the null hypothesis are used
to construct an ideal sample distribution, called expected frequencies, and then the chi-
square statistic is computed to determine how well the data (observed frequencies) fit the
hypothesis (expected frequencies). The greater the discrepancy between the data and the
hypothesis, the greater the value for chi-square and the likelihood of rejecting the null
hypothesis.
3. Students should be able to conduct a chi-square test for independence to evaluate the
relationship between two variables in the population.
For the chi-square test for independence, each individual in the sample is classified into
one category for each of two different variables. The categories of one variable form the
rows of a data matrix and the categories of the second variable form the columns. The
number of individuals in each cell of the matrix is the observed frequency for that cell.
The null hypothesis for the chi-square test for independence can be phrased two ways:
(1) there is no relationship between the two variables (they are independent), or (2) the
distribution for one variable has the same proportions for all the categories of the second
variable. Expected frequencies representing an ideal sample distribution are computed
from the null hypothesis, and the chi-square statistic is computed to determine how well
the data (observed frequencies) fit the hypothesis (expected frequencies).
4. Students should be able to evaluate the effect size (strength of relationship) for a chi-square
test of independence by computing Cohen’s w, a phi-coefficient (for a 2x2 data matrix) or
10
, Cramér’s V (for a larger data matrix).
The measures of effect size are interpreted the same as correlations and provide a
measure of the strength of the relationship.
Other Lecture Suggestions
1. For some reason, students often assume that the null hypothesis for the chi-square test for
goodness of fit always specifies equal proportions. It helps to remind them that the no-difference
hypothesis exists and that it usually produces unequal proportions across the categories. A good
classroom example compares the distribution of licensed drivers across age categories with the
distribution of automobile accidents across age categories (see problem 4 at the end of the
chapter).
2. Although the no-preference hypothesis applies to simple preference situations (like a taste test
among four brands of cola), it also applies to other areas such as questions of perceptual
discrimination. For example, people could be shown different modifications of a photograph
(eyes moved apart, mouth widened, etc.) and asked to select the most attractive. Or, infants could
be shown visual patters with different levels of complexity to determine whether they prefer to
look at one over the others.
3. When computing expected frequencies for the chi-square test for independence, point out that
the values at the bottom of each column are not free to vary but are restricted by the column
totals. Similarly, the final values in each row are restricted. If you remove the restricted values,
you are left with a matrix with (R-1) rows and (C-1) columns, hence df = (R-1)(C-1). Also note
that if students are computing the expected frequencies individually, they only need to compute a
number of values equal to the degrees of freedom: the values at the bottom of each column and at
the end of each row can be found by subtraction from the row and column totals.
4. Remind students that the expected frequencies for the chi-square test for independence should
have the same proportions across each row, not necessarily the same frequencies. To check that
they have the correct fe values, they should compute one or two proportions within a row and
compare them with the corresponding proportions computed within a different row.
11
