Overview
This lecture covers how to perform a chi-square goodness of fit test to determine if a six-sided die is fair, using observed data from 180 tosses.
Hypotheses and Setup
- Null hypothesis ((H_0)): The die is fair; observed counts equal expected counts.
- Alternative hypothesis ((H_A)): The die is not fair.
- If fair, probability for each face is (1/6), so expected count = (1/6 \times 180 = 30) for each side.
- Significance level ((\alpha)) is 0.05, as given.
Assumptions and Requirements
- Random sampling and independent measurements are assumed.
- Large sample requirement: all expected values ((30)) are greater than 5.
Test Execution: Chi-Square Goodness of Fit
- Use TI-84 calculator: Stat → Tests → Goodness of Fit Test.
- Enter observed counts in list one; expected counts in list two.
- Degrees of freedom (df) = number of categories - 1 = (6 - 1 = 5).
- Calculator output: chi-square value = 8.5; p-value = 0.1307.
Interpretation and Conclusion
- Compare p-value (0.1307) to (\alpha) (0.05).
- Since p-value > (\alpha), fail to reject the null hypothesis.
- Not enough evidence to conclude the die is unfair; differences could be due to chance in this sample size.
Key Terms & Definitions
- Null hypothesis (H₀) — The default assumption that observed data matches expected values.
- Alternative hypothesis (H₁) — The assumption that there is a significant difference between observed and expected values.
- Chi-square goodness of fit test — A statistical test to compare observed counts to expected counts in categorical data.
- Degrees of freedom (df) — Calculated as the number of categories minus one.
- Significance level (α) — The probability threshold for rejecting the null hypothesis, often set at 0.05.
- p-value — The probability of seeing the observed result (or more extreme) if the null hypothesis is true.
Action Items / Next Steps
- Practice another chi-square goodness of fit test with new data.
- Review how to input data and execute tests on your calculator.