Chi-Square Goodness of Fit (Setup) - AP Statistics

1. What is the relationship between $\chi^2$ and p-value?

   A high $\chi^2$ results in a low p-value, and vice versa. They are inversely related through the Chi-Square distribution's right tail.

2. Select the correct hypothesis test: Testing if a coin is fair.

   Chi-Square Goodness of Fit test. Tests if observed heads/tails frequencies match expected 50/50 distribution.

3. What is the shape of a Chi-Square distribution?

   Skewed to the right, especially with low degrees of freedom. The distribution becomes more symmetric as degrees of freedom increase.

4. What is the significance level typically used in hypothesis testing?

   A common significance level is $\alpha = 0.05$. This is the standard threshold for statistical significance in most tests.

5. How do you calculate degrees of freedom for a Chi-Square Goodness of Fit test?

   Degrees of freedom = number of categories - 1. Subtract 1 from categories because we lose one degree of freedom.

6. Identify the symbol $O_i$ in the Chi-Square test statistic formula.

   $O_i$ represents the observed frequency for category $i$. This is the actual count observed in each category of the data.

7. What decision do you make if the p-value is greater than the significance level?

   Fail to reject the null hypothesis. When p-value ≥ $\alpha$, insufficient evidence exists to reject the null hypothesis.

8. What assumption is needed for using the Chi-Square test?

   Random sampling and independence of observations. These ensure valid probability calculations and prevent bias in the test.

9. In what situation would you use a Chi-Square Goodness of Fit test?

   To test how well an observed distribution fits an expected distribution. This test determines if data follows a specific theoretical distribution pattern.

10. What condition must be met for expected frequencies in Chi-Square tests?

    Expected frequency in each category should be at least 5. This ensures the normal approximation to the Chi-Square distribution is valid.

11. Calculate the Chi-Square statistic: $O_i = 30, E_i = 25$ for one category.

    $\frac{(30 - 25)^2}{25} = 1.0$. Apply the formula: $(30-25)^2/25 = 25/25 = 1.0$.

12. What decision do you make if the p-value is less than the significance level?

    Reject the null hypothesis. When p-value < $\alpha$, evidence strongly contradicts the null hypothesis.

13. What does a Chi-Square Goodness of Fit test assess?

    Whether observed frequencies significantly differ from expected frequencies. Determines if the sample data matches a proposed theoretical distribution model.

14. State the formula for the Chi-Square test statistic.

    $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$. Sums squared differences between observed and expected, divided by expected.

15. Calculate degrees of freedom: 3 observed categories.

    Degrees of freedom = 2. Apply the formula: df = categories - 1 = 3 - 1 = 2.

16. Explain the purpose of a Chi-Square Goodness of Fit test briefly.

    To test if observed data fits a specified distribution. Determines if sample data follows a hypothesized probability distribution pattern.

17. What does a large Chi-Square test statistic indicate?

    A large deviation between observed and expected frequencies. Large values suggest the observed data doesn't fit the expected model well.

18. Which statistical test is used to compare observed data with a model?

    Chi-Square Goodness of Fit test. This test compares actual observations against theoretical expectations or models.

19. What is the result if $\chi^2 = 0$ in a Chi-Square test?

    Observed frequencies match expected frequencies exactly. Zero indicates perfect agreement between observed and expected frequency values.

20. What is a p-value in the context of hypothesis testing?

    The p-value is the probability of observing a test statistic as extreme as the one observed. Measures how likely the observed result is if the null hypothesis is true.

21. What does a small Chi-Square test statistic indicate?

    A small deviation between observed and expected frequencies. Small values suggest the observed data closely matches the expected model.

22. Identify the test: Testing if a die is fair based on observed roll frequencies.

    Chi-Square Goodness of Fit test. Tests if observed roll frequencies match expected equal probabilities for fairness.

23. Find the degrees of freedom: 5 categories in a Chi-Square test.

    Degrees of freedom = 4. Apply the formula: df = categories - 1 = 5 - 1 = 4.

24. What is the role of the Chi-Square distribution table?

    To find p-values and critical values for Chi-Square tests. The table provides probability values for different Chi-Square statistics and df.

25. Identify the symbol $E_i$ in the Chi-Square test statistic formula.

    $E_i$ represents the expected frequency for category $i$. This is the theoretical count predicted for each category under $H_0$.

26. Interpret $p < 0.05$ in a Chi-Square test.

    There is a statistically significant difference; reject the null hypothesis. The difference is statistically significant at the 5% level of significance.

27. What is the alternative hypothesis for a Chi-Square Goodness of Fit test?

    The observed distribution does not match the expected distribution. This states a significant difference exists between observed and expected distributions.

28. What is the mean of a Chi-Square distribution with $k$ degrees of freedom?

    Mean = $k$. The mean equals the number of degrees of freedom in the distribution.

29. What is the null hypothesis for a Chi-Square Goodness of Fit test?

    The observed distribution matches the expected distribution. This states no difference exists between observed and expected distributions.

30. What is the critical value in hypothesis testing?

    The threshold value that the test statistic must exceed to reject the null hypothesis. Determined by significance level and degrees of freedom from the distribution table.

31. What is the variance of a Chi-Square distribution with $k$ degrees of freedom?

    Variance = $2k$. The variance is always twice the degrees of freedom value.

32. Choose the type of test: Comparing a sample to a theoretical ratio.

    Chi-Square Goodness of Fit test. Tests if the sample proportions match the theoretical ratio expectations.

33. Calculate expected frequency: Total 200 trials, 4 categories, equal distribution.

    Expected frequency = 50 for each category. Divide total trials (200) by number of categories (4) for equal distribution.

34. Why might a Chi-Square test not be suitable for small sample sizes?

    Expected frequencies may be too low, violating assumptions. Small samples may not meet the minimum expected frequency requirement.

35. Which distribution does the Chi-Square test statistic follow?

    Chi-Square distribution. The test statistic follows this right-skewed distribution under the null hypothesis.

36. What is the effect of increasing categories on degrees of freedom?

    Increases degrees of freedom. More categories provide more degrees of freedom for the statistical test.

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