Instructor Manual For

Instructor Manual For Essentials of Statistics for the Behavioral Sciences, 10 th 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 10 th Edition, All other Files in The Market are Fake/Old/Wrong Edition. 1 / 4

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.

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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:

• 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.

• 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.

• 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).

• 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

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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

• 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).

• 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.

• 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.

• 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

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