Overview
This lecture covers how to use a chi-square goodness of fit test to determine if the ethnic distribution of a jury pool matches that of the community, using provided data and a significance level of 0.05.
Problem Setup
- The community's ethnic breakdown is 60% White, 20% Hispanic, 10% Asian-American, and 10% African American.
- A sample of 200 potential jurors is evaluated for matching this distribution.
- Ethnicity is the categorical variable with four categories.
Calculating Expected Values
- Expected counts: 120 White, 40 Hispanic, 20 Asian-American, 20 African American (using the given percentages of 200).
Hypotheses
- Null hypothesis (H₀): The jury pool's ethnic breakdown matches the community (Observed = Expected).
- Alternative hypothesis (H₁): The jury pool's ethnic breakdown does not match the community (Observed ≠ Expected).
Test Preparation
- Degrees of freedom = number of categories - 1 = 4 - 1 = 3.
- Alpha (significance level) is 0.05.
- Conditions: Random sample, independent measurements, all expected counts ≥ 5.
Performing the Chi-square Test
- Use the chi-square goodness of fit test.
- Observed counts: 137 White, 23 Hispanic, 18 Asian-American, 22 African American.
- Calculate (O−E)²/E for each category; sum to get the chi-square statistic.
- For TI calculators: use list functions to input data and calculate the statistic and p-value.
- The computed chi-square statistic ≈ 10.033, p-value ≈ 0.018.
Interpreting Results
- P-value (0.018) is less than alpha (0.05), so reject the null hypothesis.
- There is sufficient evidence that the jury pool's ethnic breakdown does not match the community's.
Key Terms & Definitions
- Chi-square goodness of fit test — a statistical test to compare observed categorical counts to expected counts.
- Degrees of freedom — (number of categories) minus 1; here, it equals 3.
- P-value — probability of obtaining a test statistic at least as extreme as the observed one, given the null hypothesis is true.
- Null hypothesis (H₀) — assumption that observed and expected distributions are the same.
Action Items / Next Steps
- Practice identifying when to use the chi-square goodness of fit test.
- Complete homework 10.2.2 for further understanding.