Difference of Two Population Proportions (Test) - AP Statistics

1. What is the critical value for a one-tailed test at 5% significance level?

   Approximately $1.645$ or $-1.645$. One-tailed critical values at 5% significance level.

2. What is the formula for calculating a confidence interval for $p_1-p_2$?

   $(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$. Uses separate standard errors for confidence interval.

3. Identify the alternative hypothesis for a one-tailed test where $p_1 < p_2$.

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

4. State the null hypothesis for testing the difference between two proportions.

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

5. Determine the decision rule for rejecting $H_0$ in a z-test.

   Reject $H_0$ if $|z| > z_{critical}$. Reject when test statistic exceeds critical value.

6. Identify the pooled sample proportion formula in a two-proportion test.

   $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$. Combines successes from both samples over total sample size.

7. Determine the degrees of freedom used in a two-proportion z-test.

   Not applicable; z-test uses standard normal distribution. Z-test uses normal distribution, not t-distribution.

8. Identify the null hypothesis if $p_1 = p_2$ is assumed.

   $H_0: p_1 - p_2 = 0$. Null hypothesis assumes equal population proportions.

9. Identify the statistical test for $n_1 = 30$, $n_2 = 40$, $x_1 = 15$, $x_2 = 20$.

   Two-proportion z-test. Sample sizes meet conditions for normal approximation.

10. What does a two-proportion z-test evaluate?

    Difference between two population proportions. Tests whether two population proportions are equal.

11. Calculate $\hat{p}_1$ with $x_1 = 40$, $n_1 = 120$.

    $\hat{p}_1 = 0.333$. Sample proportion: $40/120 = 0.333$.

12. Find $\hat{p}$ using $x_1 = 30$, $x_2 = 40$, $n_1 = 100$, $n_2 = 100$.

    $\hat{p} = 0.35$. Pooled proportion: $(30+40)/(100+100) = 0.35$.

13. Evaluate if $H_0$ is rejected if $z = 2.5$ and $z_{critical} = 1.96$.

    Yes, reject $H_0$. Test statistic exceeds critical value, so reject null.

14. Calculate the pooled proportion for $x_1 = 25$, $x_2 = 35$, $n_1 = 50$, $n_2 = 70$.

    $\hat{p} = 0.5$. Pooled proportion: $(25+35)/(50+70) = 0.5$.

15. Determine if $H_0$ is rejected with $p$-value = 0.03, $\alpha = 0.05$.

    Yes, reject $H_0$. P-value less than significance level, reject null.

16. State the alternative hypothesis for testing the difference between two proportions.

    $H_a: p_1 - p_2 \neq 0$ (or $>0$ or $<0$). Tests for significant difference in either direction or one-sided.

17. Identify the formula for calculating $\hat{p}_1$ in a sample.

    $\hat{p}_1 = \frac{x_1}{n_1}$. Sample proportion equals successes divided by sample size.

18. Find the critical value for a two-tailed test at 5% significance level.

    Approximately $\pm 1.96$. Standard normal distribution critical values at $\alpha = 0.05$.

19. What is the condition for normality in a two-proportion z-test?

    Sample sizes are large enough for a normal approximation. Large sample sizes ensure sampling distribution normality.

20. Identify the formula for calculating $\hat{p}_2$ in a sample.

    $\hat{p}_2 = \frac{x_2}{n_2}$. Second sample proportion formula for group 2.

21. What is the purpose of a two-proportion z-test?

    To compare two population proportions. Determines if two population proportions differ significantly.

22. What is the formula for the test statistic in a two-proportion z-test?

    $z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$. Uses pooled proportion for standard error under null hypothesis.

23. What is the assumption regarding independence in a two-proportion z-test?

    Samples must be independent. Two samples must be selected independently.

24. What is the condition for sample size in a two-proportion z-test?

    Each group should have at least 10 successes and 10 failures. Ensures normal approximation is valid for both samples.

25. Identify the significance level if $z = 1.96$ at a two-tailed test.

    $\alpha = 0.05$. Critical value $1.96$ corresponds to $\alpha = 0.05$ two-tailed.

26. Find the $z$-score if $\hat{p}_1 = 0.6$, $\hat{p}_2 = 0.4$, $n_1 = n_2 = 50$.

    $z = 2.83$. Using pooled $\hat{p} = 0.5$ in standard error calculation.

27. Calculate $\hat{p}$ if $x_1 = 10$, $x_2 = 15$, $n_1 = 50$, $n_2 = 60$.

    $\hat{p} = 0.227$. Pooled proportion: $(10+15)/(50+60) = 0.227$.

28. Which test is used for large sample sizes in comparing two proportions?

    Two-proportion z-test. Normal approximation valid for large samples.

29. Identify the alternative hypothesis for a one-tailed test where $p_1 > p_2$.

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

30. State the meaning of a p-value in hypothesis testing.

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

31. Find the critical value for a 99% confidence interval.

    Approximately $\pm 2.576$. 99% confidence level critical values from normal distribution.

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

    Standard error decreases. Larger samples reduce sampling variability.

33. What does a z-test assume about the sampling distribution?

    It is approximately normal. Central Limit Theorem ensures normality for large samples.

34. Find the $z$-score for $\hat{p}_1 = 0.4$, $\hat{p}_2 = 0.5$, $n_1 = n_2 = 100$.

    $z = -2.04$. Using pooled $\hat{p} = 0.45$ and standard error calculation.

35. Calculate $\hat{p}_2$ with $x_2 = 50$, $n_2 = 150$.

    $\hat{p}_2 = 0.333$. Sample proportion: $50/150 = 0.333$.

36. What is the role of the standard error in a two-proportion z-test?

    Measures the variability of the sampling distribution. Quantifies uncertainty in the difference estimate.

37. Calculate the standard error for a difference in sample proportions.

    $\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}$. Measures variability of difference in sample proportions.

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