## Mood’s test :

Mood’s test, also known as the chi-squared test for independence, is a statistical test used to determine if there is a significant association between two categorical variables. It is commonly used in research to examine the relationship between a dependent variable and one or more independent variables.

One example of using Mood’s test is to examine the relationship between gender and voting behavior. In this case, the dependent variable is voting behavior (e.g. Republican, Democrat, Independent), and the independent variable is gender (male, female). The null hypothesis would be that there is no relationship between gender and voting behavior, while the alternative hypothesis is that there is a significant relationship.

To conduct the test, researchers would collect data on a sample of individuals and their voting behavior and gender. They would then create a contingency table, which shows the frequency of each combination of voting behavior and gender. For example, if there are 100 individuals in the sample, 40 of whom are male and 60 of whom are female, the contingency table may look like this:

Voting behavior Male Female

Republican 15 25

Democrat 10 35

Independent 15 10

Next, the researchers would calculate the expected frequencies for each combination of voting behavior and gender if the null hypothesis were true. In this example, if there is no relationship between gender and voting behavior, we would expect the number of males and females in each voting behavior category to be proportional to the overall sample. In this case, the expected frequencies would be:

Voting behavior Male Female

Republican 20 30

Democrat 15 45

Independent 5 15

Next, the researchers would calculate the chi-squared statistic, which is a measure of the discrepancy between the observed and expected frequencies. This is done by summing the squared differences between the observed and expected frequencies, divided by the expected frequencies, for each combination of voting behavior and gender. In this example, the chi-squared statistic would be:

Chi-squared = ((15-20)^2 / 20) + ((10-15)^2 / 15) + ((15-5)^2 / 5) + ((25-30)^2 / 30) + ((35-45)^2 / 45) + ((10-15)^2 / 15) = 9.33

Finally, the researchers would compare the calculated chi-squared statistic to a critical value from a chi-squared distribution table, based on the number of degrees of freedom (which is equal to the number of categories of the dependent variable minus one). If the calculated chi-squared statistic is greater than the critical value, then the null hypothesis is rejected, and there is a significant relationship between gender and voting behavior. In this example, if the sample size is 100 and the number of categories of the dependent variable is 3, the critical value would be 7.815. Since the calculated chi-squared statistic of 9.33 is greater than the critical value, the null hypothesis is rejected, and there is a significant relationship between gender and voting behavior.