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

1. What is the consequence if expected frequencies are less than 5?

   Chi-Square test may not be valid. Violates assumptions needed for valid statistical inference.

2. What is the standard level of significance used in hypothesis testing?

   0.05. Most commonly used threshold for statistical significance.

3. Identify the primary condition for using the Chi-Square test.

   Data must be categorical. Chi-square tests require frequency counts of categories.

4. What is the effect of increasing the sample size on the Chi-Square test?

   Increases the power of the test. Larger samples better detect true differences from null.

5. Which type of data is inappropriate for a Chi-Square test?

   Continuous data. Chi-square requires categorical frequency data only.

6. Find the degrees of freedom for a test with 7 categories.

   Degrees of freedom = 6. Formula: $df = k - 1$ where $k$ is categories.

7. Choose the appropriate test: 'Comparing the distribution of a sample to a theoretical distribution.'

   Chi-Square Goodness of Fit test. Tests if sample matches theoretical distribution pattern.

8. Identify the degrees of freedom for a Chi-Square Goodness of Fit test with 5 categories.

   Degrees of freedom = 4. Formula: $df = k - 1$ where $k$ is number of categories.

9. Identify the assumption about sample size in a Chi-Square test.

   Sample size should be large enough to ensure expected frequencies are at least 5. Prevents invalid approximation to chi-square distribution.

10. What is the relationship between Chi-Square statistic and p-value?

    Higher Chi-Square statistic typically results in a lower p-value. Larger deviations produce higher statistics and lower p-values.

11. What is the purpose of a Chi-Square Test for Goodness of Fit?

    To determine if observed frequencies match expected frequencies. Tests if data follows a specific distribution pattern.

12. What are the two main types of Chi-Square tests?

    Goodness of Fit and Test of Independence. Both test distribution assumptions but in different ways.

13. Identify the correct statement: 'Chi-Square test is for categorical or numerical data?'

    Categorical data. Chi-square applies to frequency counts of categories.

14. What is the critical value in a Chi-Square test used for?

    To determine whether to reject the null hypothesis. Compare with test statistic to make statistical decision.

15. Identify the effect of a larger Chi-Square statistic value.

    Greater evidence against the null hypothesis. Higher values indicate greater deviation from expected.

16. What is the formula for calculating degrees of freedom in a Chi-Square Goodness of Fit test?

    $df = k - 1$. Where $k$ is the number of categories being tested.

17. What is the primary goal of calculating the expected frequencies?

    To compare against observed frequencies. Baseline for measuring deviation from null hypothesis.

18. Which distribution is used for the Chi-Square test?

    Chi-Square distribution. Theoretical distribution for chi-square test statistic.

19. How do you calculate expected frequency for each category?

    $E_i = \frac{\text{Total Observed} \times \text{Proportion}}{1}$. Multiply total sample size by theoretical probability.

20. What is the null hypothesis in a Chi-Square Goodness of Fit test?

    The observed frequencies match the expected frequencies. Assumes the theoretical distribution is correct.

21. What does a p-value greater than 0.05 suggest in a Chi-Square test?

    Fail to reject the null hypothesis. Insufficient evidence to reject null hypothesis.

22. What is the shape of the Chi-Square distribution?

    Right-skewed. Chi-square values are always non-negative with long right tail.

23. What does a low p-value indicate in a Chi-Square test?

    Strong evidence against the null hypothesis. Small p-values suggest data unlikely under null hypothesis.

24. What does $E_i$ represent in the Chi-Square formula?

    $E_i$ represents the expected frequency. The subscript $i$ denotes each category under null hypothesis.

25. State the formula for the Chi-Square statistic.

    $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$. Compares observed and expected frequencies for each category.

26. What happens to the Chi-Square distribution as degrees of freedom increase?

    It becomes more symmetric. Approaches normal distribution as degrees of freedom increase.

27. Find the critical value for a Chi-Square test with 3 degrees of freedom at 0.05 significance.

    7.815. From chi-square table with $\alpha = 0.05$ and $df = 3$.

28. Find the degrees of freedom for a test with 10 categories.

    Degrees of freedom = 9. Formula: $df = k - 1$ where $k$ is categories.

29. How is the p-value related to the Chi-Square statistic?

    P-value indicates the probability of observing a test statistic as extreme. Area under chi-square curve beyond observed statistic.

30. What is the assumption about the expected frequency in a Chi-Square test?

    Expected frequency should be at least 5. Ensures valid approximation to chi-square distribution.

31. What is the alternative hypothesis in a Chi-Square Goodness of Fit test?

    The observed frequencies do not match the expected frequencies. At least one category differs from expected distribution.

32. What does $O_i$ represent in the Chi-Square formula?

    $O_i$ represents the observed frequency. The subscript $i$ denotes each category in the data set.

33. Which hypothesis do you reject if the Chi-Square statistic exceeds the critical value?

    Null hypothesis. Reject when evidence strongly contradicts null assumption.

34. Which table do you use to find Chi-Square critical values?

    Chi-Square distribution table. Provides critical values based on degrees of freedom.

35. What criterion determines the rejection of the null hypothesis in a Chi-Square test?

    Chi-Square statistic > critical value. Decision rule for hypothesis testing with chi-square.

36. What is the significance level commonly used in a Chi-Square test?

    0.05. Standard alpha level for hypothesis testing.

37. What does a significant Chi-Square result indicate?

    Observed frequencies differ from expected frequencies. Provides evidence against null hypothesis of equal distribution.

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