Chi-Squared Test Examples

Introduction to the Chi-Squared Test

The Chi-Squared Test is a non-parametric statistical test used to determine if there is a significant association between categorical variables. It evaluates how well the observed frequencies of a dataset match the expected frequencies under the null hypothesis. This test is widely used in various fields, including biology, marketing, and social sciences. Below are three practical examples that illustrate the application of the Chi-Squared Test.

Example 1: Effect of Diet on Weight Loss

In a clinical study, researchers want to evaluate whether three different diets (A, B, and C) lead to different weight loss outcomes. Participants are randomly assigned to one of the three diets and after 12 weeks, their weight loss is recorded as a successful outcome (lost more than 5% of body weight) or unsuccessful outcome (lost less than 5% of body weight).

The data collected is as follows:

To determine if there is a significant difference in the success rates of the diets, a Chi-Squared Test is performed.

1. Calculate the expected frequencies for each cell based on the total proportions.

2. Use the formula for Chi-Squared:

   \[ \chi^2 = \sum \frac{(O - E)^2}{E} \]
   where O is the observed frequency and E is the expected frequency.

3. Determine the degrees of freedom, which is (rows - 1) * (columns - 1).

From this analysis, the researchers can conclude whether the diet type significantly affects weight loss outcomes.

Notes

• Variations can include different diet types or additional outcome measures (like BMI changes).

Example 2: Surveying Preferences for Ice Cream Flavors

A local ice cream shop wants to understand customer preferences for three flavors: Chocolate, Vanilla, and Strawberry. They conduct a survey during a weekend, asking 100 customers about their favorite flavor.

The survey results are as follows:

To analyze if the customers’ preferences are evenly distributed among the flavors, a Chi-Squared Test is conducted.

1. The expected frequency for each flavor, assuming equal preference, would be 33.33 (100 customers/3 flavors).
2. The Chi-Squared statistic is calculated using the observed and expected frequencies.
3. The degrees of freedom in this case would be 2 (3 flavors - 1).

The shop can then assess whether their ice cream flavors are equally liked or if further marketing is needed.

Notes

• This test can be expanded by including more flavors or customer demographics.

Example 3: Relationship Between Education Level and Employment Status

A researcher wants to investigate the relationship between education levels (High School, Bachelor’s, Master’s) and employment status (Employed, Unemployed). They collect data from a sample of 300 individuals.

The results are summarized in the following contingency table:

To examine if education level affects employment status, a Chi-Squared Test will be performed.

1. Calculate the expected frequencies for each cell.
2. Apply the Chi-Squared formula to compute the test statistic.
3. Determine the degrees of freedom, which is (3 - 1) * (2 - 1) = 2.

The results will help the researcher understand the impact of education on employment outcomes.

Notes

• Further analysis could include different employment sectors or age groups to make the study more comprehensive.