Think of a situation like the thinking-suppression device example. Come up with
a plausible scenario, with real-life data, using an observable that is binomial, and
which you can measure in at least two di�erent groups. For example: you might
ask if somebody is a business or non-business major and see if they are a member
of a soroity or fraternity. Then do the classical z-test (or �2) test. Show me the
data and the results from the test.
Sure! Let's say we want to investigate if there is a relationship between the type of major a student is pursuing (Business or Non-business) and their involvement in extracurricular activities (Member of a fraternity/sorority or not). We can collect data from a sample of students on their major and extracurricular membership status.
Here's an example scenario with hypothetical data:
We surveyed 100 students, 50 of whom are studying Business (Group A) and 50 studying Non-business (Group B). Out of the Business major students, 30 are members of a fraternity/sorority, while 15 of the Non-business major students are members.
Based on this data, we can set up the following hypothesis for testing:
Null Hypothesis (H0): There is no association between the type of major and membership in a fraternity/sorority.
Alternative Hypothesis (H1): There is an association between the type of major and membership in a fraternity/sorority.
To conduct a hypothesis test, we can use the chi-squared (χ2) test for independence. This test assesses if there is a statistically significant relationship between two categorical variables.
To calculate the chi-square statistic, we create a contingency table that cross-tabulates the major type and membership status:
Member of Fraternity/Sorority Not a Member Total
Business 30 20 50
Non-business 15 35 50
Total 45 55 100
Next, we calculate the expected values under the assumption of independence between the variables. The expected value is obtained by multiplying the row total by the column total and dividing by the grand total:
Member of Fraternity/Sorority Not a Member Total
Business 22.5 27.5 50
Non-business 22.5 27.5 50
Total 45 55 100
Now, we can calculate the chi-square statistic using the formula:
χ2 = ∑ [(Observed - Expected)2 / Expected]
In this example, the calculation would involve summing the contributions from all the cells in the table.
Once we have the chi-square value, we can look up its corresponding p-value in a chi-square distribution table (or use statistical software). If the p-value is less than our chosen significance level (e.g., 0.05), we reject the null hypothesis and conclude that there is a significant association between the type of major and membership in a fraternity/sorority.
I hope this explanation helps you understand how to set up and conduct a chi-square test for independence using real-life data!