Question

In: Statistics and Probability

When and how do you use a chi-square distribution to test if two variables are independent?...


When and how do you use a chi-square distribution to test if two variables are independent? What is an example of how to use the contingency table to find expected frequencies?

Solutions

Expert Solution

The Chi-Square test of independence is used to determine if there is a significant relationship between two nominal (categorical) variables. The frequency of each category for one nominal variable is compared across the categories of the second nominal variable. The data can be displayed in a contingency table where each row represents a category for one variable and each column represents a category for the other variable. For example, say a researcher wants to examine the relationship between gender (male vs. female) and empathy (high vs. low). The chi-square test of independence can be used to examine this relationship. The null hypothesis for this test is that there is no relationship between gender and empathy. The alternative hypothesis is that there is a relationship between gender and empathy (e.g. there are more high-empathy females than high-empathy males).

Let us consider the following example :

Suppose we have the observed counts we need to compute the expected counts under the null hypothesis that the two categorical variables are independent. This is done using the marginal totals and overall total to compute expected counts for each cell of the table. In words, to find the expected count for each cell in the table we take multiply the marginal row and column totals for that cell and divide by the overall total. Formulaically for each cell this is:

To conduct this test we compute a Chi-square test statistic where we compare each cell's observed count to it's respective expected count. This Chi-square test statistic is calculated as follows:

We make our decision by either comparing the value of the test statistic to a critical value (rejection region approach), or by finding the probability of getting this test statistic value or one more extreme (p-value approach). The critical value for our Chi-square test is χ2alpha with degree of freedom = (r - 1) (c - 1), while the p-value is found by P(χ2alpha > χ2∗) with degrees of freedom = (r - 1)(c - 1)

So, in this problem our hypothesis are :

Null hypothesis : Party Affiliation and Opinion on Tax Reform are independent.

Alternative hypothesis : Party Affiliation and Opinion on Tax Reform are not independent.

Now, Tabulated χ2alpha = 7.377

Since,  χ2∗ = 22.152 > 7.377 (χ2alpha) , we reject null hypothesis and conclude that Party Affiliation and Opinion on Tax Reform are not independent.


Related Solutions

How do you know when to conduct a t-test or a chi-square test?
How do you know when to conduct a t-test or a chi-square test?
4.) When would you use the normal distribution? T-distribution? Chi-Square distribution?
4.) When would you use the normal distribution? T-distribution? Chi-Square distribution?
You have been using the chi-square test of independence to test for association between two variables....
You have been using the chi-square test of independence to test for association between two variables. 1) Give an example of a pair of variables for which you would expect to find no association. 2) Give an example of a pair of variables for which you would expect to find association, but not a causal relationship. 3) Finally, give an example of a pair of variables for which you would expect to find a causal relationship.
3. How many qualitative variables is the chi-square test of a contingency table a test of...
3. How many qualitative variables is the chi-square test of a contingency table a test of independence for? 11.3 Chi-Square Test for Independence 4. How many expected cell frequencies satisfy subsequent assumptions, so that the chi-square test of a contingency table is valid? 11.3 Chi-Square Test for Independence Page 368 Test Review Questions Chapter 12 1. What is the name of the variable that's used to predict another variable? 12.1 The Simple Linear Regression Model 2. What is the “standard...
1.a). When do researchers use ANOVA vs. Chi-square test? Indicate two main characteristics of ANOVA and...
1.a). When do researchers use ANOVA vs. Chi-square test? Indicate two main characteristics of ANOVA and Chi-square tests (two for each test). 1.b). What are some of the major limitations of both ANOVA and Chi-square tests? (at least two for each)
What is the Chi-Square test? When and how are they useful? Explain.
What is the Chi-Square test? When and how are they useful? Explain.
When we run a chi-square test of independence on a 2x2 table, the resulting Chi-square test...
When we run a chi-square test of independence on a 2x2 table, the resulting Chi-square test statistic would be equal to the square of the Z-test statistic from the Z-test of two independent proportions. True or false
Explain the Chi-Square test and when it is appropriate.
Explain the Chi-Square test and when it is appropriate.
Chi Square Test We will now use Excel to run an example of a chi square...
Chi Square Test We will now use Excel to run an example of a chi square test. Chi square test is checking the independence of two variables. Our example will test if taking hormonal pills and being overweight are related. We will test the independence on 200 random patients. Thus, N=200. They will be divided first into two groups, those who take hormonal pills and those who do not. Second, they will be divided into three groups based on weight,...
When would you use ANOVA instead of a chi-square test? Is it possible to establish through...
When would you use ANOVA instead of a chi-square test? Is it possible to establish through a regression analysis if a variable is a dependent or an independent variable? Why or why not?
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT