Chi-Square Goodness of Fit Test Examples

Explore practical examples of chi-square goodness of fit tests across various contexts.
By Jamie

Understanding the Chi-Square Goodness of Fit Test

The Chi-Square Goodness of Fit Test is a statistical method used to determine if a sample distribution matches an expected distribution. It helps researchers assess how well their observed data fit a specific theoretical model. This test is particularly useful in various fields, including biology, marketing, and social sciences. Below are three diverse, practical examples of using the chi-square goodness of fit test.

Example 1: Analyzing Color Distribution in M&Ms

In a study to investigate whether the color distribution of M&Ms matches the company’s claimed proportions, a researcher collects data from a random sample of M&Ms. The expected proportions for each color are as follows:

  • Red: 20%
  • Yellow: 20%
  • Green: 20%
  • Blue: 20%
  • Brown: 20%

After counting the colors in a sample of 100 M&Ms, the researcher finds:

  • Red: 15
  • Yellow: 25
  • Green: 20
  • Blue: 30
  • Brown: 10

To perform the chi-square goodness of fit test, the researcher calculates the chi-square statistic using the formula:

Chi-Square Formula

Where:

  • Observed values (O) are the counted M&Ms’ colors.
  • Expected values (E) are based on the claimed proportions.

Calculating the expected counts for each color:

  • Red: 20
  • Yellow: 20
  • Green: 20
  • Blue: 20
  • Brown: 20

After calculating the test statistic and comparing it to the critical value, the researcher concludes whether the distribution significantly differs from the expected proportions.

Notes

  • This example highlights the application of chi-square tests in quality control and marketing research.
  • A variation could involve testing color distribution in a different candy brand or product.

Example 2: Surveying Favorite Ice Cream Flavors

A local ice cream shop wants to understand customer preferences for flavors. They conduct a survey asking patrons to choose their favorite flavor from a list of five options:

  • Vanilla
  • Chocolate
  • Strawberry
  • Mint
  • Cookie Dough

The expected preference distribution based on past sales data is:

  • Vanilla: 30%
  • Chocolate: 25%
  • Strawberry: 20%
  • Mint: 15%
  • Cookie Dough: 10%

From 200 survey responses, the observed counts are:

  • Vanilla: 50
  • Chocolate: 60
  • Strawberry: 30
  • Mint: 40
  • Cookie Dough: 20

Using the chi-square goodness of fit test, the shop owner calculates the chi-square statistic to see if the observed preferences significantly differ from the expected distribution.

Notes

  • This example can be extended by including demographic variables to see if preferences differ among age groups.
  • The survey may also include additional flavors to analyze emerging trends.

Example 3: Classroom Gender Ratio Analysis

A teacher wishes to analyze the gender ratio in their classroom against the expected ratio in the school district. The expected ratio is:

  • Male: 50%
  • Female: 50%

In a class of 30 students, the observed counts are:

  • Male: 20
  • Female: 10

To perform the chi-square goodness of fit test, the teacher compares the observed counts to the expected values:

  • Expected counts for Male: 15
  • Expected counts for Female: 15

The chi-square statistic is calculated to determine if the observed gender distribution significantly deviates from the expected equal distribution.

Notes

  • This example illustrates the chi-square test’s application in educational settings.
  • Variations could include analyzing gender ratios across different grades or subjects.

By utilizing these practical examples, readers should have a clearer understanding of how to apply the chi-square goodness of fit test in various real-world scenarios.