Chi-Square Test of Goodness of Fit vs. Chi-Square Test of Independence

The chi-square test is a statistical test used to determine if there is a significant relationship between two categorical variables. However, there are different scenarios in which either the chi-square test of goodness of fit or the chi-square test of independence would be more appropriate.

Chi-Square Test of Goodness of Fit

The chi-square test of goodness of fit is used when we want to determine if an observed frequency distribution differs significantly from an expected frequency distribution. This test is typically employed when we have one categorical variable with multiple categories and we want to compare the observed frequencies in each category with the expected frequencies. For example, let’s say we have a sample of 200 people and we want to determine if their political affiliations differ significantly from what would be expected based on the general population. In this case, our null hypothesis would be that the observed frequencies of political affiliations are equal to the expected frequencies. The alternative hypothesis would be that there is a significant difference between the observed and expected frequencies.

Chi-Square Test of Independence

On the other hand, the chi-square test of independence is used when we want to determine if there is a relationship between two categorical variables. This test is employed when we have two categorical variables and we want to assess whether changes in one variable are associated with changes in the other variable. For instance, suppose we are conducting a study on gender and voting preference. We want to investigate if there is a relationship between gender (male/female) and voting preference (Republican/Democrat/Independent). The research question could be: “Is there a significant association between gender and voting preference?”

Steps to Computing a Chi-Square Test

To compute a chi-square test, follow these steps:

1. State the null hypothesis (H0) and the alternative hypothesis (Ha).
2. Set the significance level (alpha) for the test.
3. Collect data and create a contingency table, which displays the observed frequencies for each combination of categories.
4. Calculate the expected frequencies for each cell in the contingency table. The expected frequency is calculated by multiplying the row total by the column total and dividing it by the overall sample size.
5. Calculate the chi-square test statistic using the formula: X2 = Σ((O-E)2 / E), where O is the observed frequency and E is the expected frequency.
6. Determine the degrees of freedom (df) for the test, which is calculated as (r-1)(c-1), where r is the number of rows and c is the number of columns in the contingency table.
7. Look up the critical value from the chi-square distribution table using the degrees of freedom and significance level.
8. Compare the calculated chi-square test statistic with the critical value. If the calculated value exceeds the critical value, reject the null hypothesis; otherwise, fail to reject the null hypothesis.
9. Report the results, including the chi-square test statistic, degrees of freedom, p-value, and conclusion about the null hypothesis.

Reporting for Chi-Square Test of Independence

When reporting the results of a chi-square test of independence, it is important to include the following information:

1. The chi-square test statistic value (X2).
2. The degrees of freedom (df).
3. The p-value associated with the test.
4. A conclusion about the null hypothesis, indicating whether it should be rejected or failed to be rejected.

For example, “A chi-square test of independence was conducted to examine the relationship between gender and voting preference. The chi-square test statistic was X2 = 16.32 with 2 degrees of freedom, p < 0.001. Therefore, we reject the null hypothesis and conclude that there is a significant association between gender and voting preference.” In summary, the chi-square test of goodness of fit is used to compare observed and expected frequencies within a single categorical variable, while the chi-square test of independence is employed to assess the relationship between two categorical variables. The steps for computing a chi-square test involve stating hypotheses, collecting data, calculating the test statistic, determining degrees of freedom, comparing with critical values, and reporting the results.

Sample Answer