Difference of Two Population Proportions (Setup) - AP Statistics

1. What is the alternative hypothesis for a two-tailed test of two proportions?

   $H_a: p_1 - p_2 \neq 0$. Tests if proportions differ in either direction (two-sided test).

2. What is the critical region in hypothesis testing?

   Area in tails of distribution where $H_0$ is rejected. Region where test statistic leads to $H_0$ rejection.

3. What is the relationship between Type I and Type II errors?

   Reducing one increases the other, given a fixed sample size. Trade-off relationship between the two types of errors.

4. If $p$-value = 0.03 and $\alpha = 0.05$, what is the decision regarding $H_0$?

   Reject $H_0$. P-value is less than significance level.

5. What does a $p$-value less than $\alpha$ indicate in hypothesis testing?

   Strong evidence against $H_0$. Statistically significant result supporting the alternative hypothesis.

6. What are critical values in hypothesis testing?

   Thresholds for deciding whether to reject $H_0$. Z-scores that define the rejection region boundaries.

7. How do you calculate the test statistic for two sample proportions?

   Use the Z formula with pooled standard error. Apply the standardized test statistic formula.

8. What are the assumptions for testing the difference of two proportions?

   Random samples, independent groups, large sample size. Required conditions for valid statistical inference.

9. What happens to the power of a test if the sample size is increased?

   Power increases. Larger samples improve the test's ability to detect differences.

10. What does a $p$-value greater than $\alpha$ indicate in hypothesis testing?

    Insufficient evidence to reject $H_0$. Not enough evidence to conclude a significant difference.

11. What is the critical value for a 95% confidence level in a Z-test?

    1.96. Boundary value for 95% confidence in two-tailed tests.

12. What condition must be met for the sample size in a test of two proportions?

    Both $n_1$ and $n_2$ should be sufficiently large. Ensures normal approximation is valid.

13. What is the formula for the test statistic in a two-proportion Z-test?

    $Z = \frac{(\hat{p}_1 - \hat{p}_2)}{SE}$. Standardizes the difference using pooled standard error.

14. What is the alternative hypothesis for a right-tailed test of two proportions?

    $H_a: p_1 - p_2 > 0$. Tests if the first proportion is greater than the second.

15. State the formula for the standard error of the difference of two sample proportions.

    $SE = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$. Measures variability of the difference between sample proportions.

16. What is the significance level commonly used in hypothesis tests?

    $\alpha = 0.05$. Standard threshold for statistical significance.

17. What is the impact of a higher significance level on Type I error probability?

    Increases Type I error probability. Higher $\alpha$ means greater chance of false positive.

18. Define a Type I error in the context of hypothesis testing.

    Rejecting $H_0$ when it is true. False positive: concluding difference exists when it doesn't.

19. Which condition must be checked for normality in two-proportion tests?

    Both $n_1\hat{p}_1$, $n_1(1-\hat{p}_1)$, $n_2\hat{p}_2$, $n_2(1-\hat{p}_2) > 5$. Ensures sufficient data for normal approximation validity.

20. If $p$-value = 0.07 and $\alpha = 0.05$, what is the decision regarding $H_0$?

    Fail to reject $H_0$. P-value exceeds the significance level.

21. What is the effect of increasing the sample size on the standard error?

    Decreases the standard error. Standard error decreases as sample size increases.

22. Calculate $\hat{p}$ if $x_1 = 30$, $n_1 = 100$, $x_2 = 40$, $n_2 = 150$.

    $\hat{p} = \frac{70}{250}$. Total successes divided by total sample size.

23. How is the power of a test defined?

    Probability of rejecting $H_0$ when $H_a$ is true. Ability to detect a true difference when it exists.

24. What is the decision rule for $p$-value in hypothesis testing?

    Reject $H_0$ if $p$-value $< \alpha$. Standard criterion for determining statistical significance.

25. What is the purpose of setting a significance level in hypothesis testing?

    To determine the threshold for rejecting $H_0$. Sets the cutoff for statistical significance decisions.

26. What is the alternative hypothesis for a left-tailed test of two proportions?

    $H_a: p_1 - p_2 < 0$. Tests if the first proportion is less than the second.

27. What does it mean if the confidence interval for $p_1 - p_2$ includes 0?

    Fail to reject $H_0$. Zero difference is plausible, so no significant difference.

28. How do you denote the difference between two population proportions?

    $p_1 - p_2$. Parameter of interest in the hypothesis test.

29. What is the null hypothesis for testing the difference of two proportions?

    $H_0: p_1 - p_2 = 0$. Assumes no difference between the two population proportions.

30. What role does the sample size play in hypothesis testing?

    Larger samples lead to more reliable results. Larger samples reduce sampling variability and increase precision.

31. Define a Type II error in hypothesis testing.

    Failing to reject $H_0$ when $H_a$ is true. False negative: missing a real difference between proportions.

32. What does $p$-value represent in hypothesis testing?

    Probability of observing data as extreme as the sample given $H_0$. Measures strength of evidence against the null hypothesis.

33. Identify the pooled proportion formula for a test of two population proportions.

    $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$. Combines both samples to estimate common proportion under $H_0$.

34. What is the pooled standard error formula for two proportions?

    $SE_p = \sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}$. Uses pooled proportion to calculate variability under $H_0$.

35. What is meant by 'independent groups' in the context of hypothesis testing?

    Samples are not related or paired. No connection between observations in different groups.

36. What are the symbols for sample proportions in hypothesis testing?

    $\hat{p}_1$ and $\hat{p}_2$. Observed proportions from each sample group.

37. Which test statistic is used for comparing two population proportions?

    Z-test. Uses normal distribution for large samples.

38. What condition is necessary for using a Z-test for two proportions?

    Sample sizes must be large enough for normal approximation. Requires at least 5 successes and failures in each group.

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