AP Statistics Flashcards: Difference Of Two Population Proportions Test

What this deck covers

This deck focuses on Difference Of Two Population Proportions Test, giving you a quick way to review the definitions, rules, and examples that matter most for AP Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

Flashcard 2: What does a z-test assume about the sampling distribution?

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

Flashcard 3: What is the formula for calculating a confidence interval for $p_1-p_2$?

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

Flashcard 4: Identify the formula for calculating $\hat{p}_1$ in a sample.

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

Flashcard 5: Identify the pooled sample proportion formula in a two-proportion test.

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

Flashcard 6: Identify the alternative hypothesis for a one-tailed test where $p_1 < p_2$.

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

Flashcard 7: Evaluate if $H_0$ is rejected if $z = 2.5$ and $z_{critical} = 1.96$.

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

Flashcard 8: State the null hypothesis for testing the difference between two proportions.

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

Flashcard 9: Determine the decision rule for rejecting $H_0$ in a z-test.

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

Flashcard 10: Identify the pooled sample proportion formula in a two-proportion test.

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

Flashcard 11: Determine the degrees of freedom used in a two-proportion z-test.

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

Flashcard 12: What does a two-proportion z-test evaluate?

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

Flashcard 13: Identify the null hypothesis if $p_1 = p_2$ is assumed.

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

Flashcard 14: Identify the statistical test for $n_1 = 30$, $n_2 = 40$, $x_1 = 15$, $x_2 = 20$.

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

Flashcard 15: State the alternative hypothesis for testing the difference between two proportions.

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

Flashcard 16: What is the role of the standard error in a two-proportion z-test?

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

Flashcard 17: What does a two-proportion z-test evaluate?

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

Flashcard 18: What is the formula for calculating a confidence interval for $p_1-p_2$?

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

Flashcard 19: Calculate $\hat{p}_1$ with $x_1 = 40$, $n_1 = 120$.

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

Flashcard 20: Determine the decision rule for rejecting $H_0$ in a z-test.

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

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

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

Flashcard 22: Calculate $\hat{p}$ if $x_1 = 10$, $x_2 = 15$, $n_1 = 50$, $n_2 = 60$.

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

Flashcard 23: Find $\hat{p}$ using $x_1 = 30$, $x_2 = 40$, $n_1 = 100$, $n_2 = 100$.

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

Flashcard 24: Find $\hat{p}$ using $x_1 = 30$, $x_2 = 40$, $n_1 = 100$, $n_2 = 100$.

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

Flashcard 25: What is the effect of increasing sample size on the standard error?

Answer: Standard error decreases. Larger samples reduce sampling variability.

Flashcard 26: Determine the degrees of freedom used in a two-proportion z-test.

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

Flashcard 27: Identify the null hypothesis if $p_1 = p_2$ is assumed.

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

Flashcard 28: Evaluate if $H_0$ is rejected if $z = 2.5$ and $z_{critical} = 1.96$.

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

Flashcard 29: Calculate the pooled proportion for $x_1 = 25$, $x_2 = 35$, $n_1 = 50$, $n_2 = 70$.

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

Flashcard 30: Find the $z$-score if $\hat{p}_1 = 0.6$, $\hat{p}_2 = 0.4$, $n_1 = n_2 = 50$.

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

Flashcard 31: Determine if $H_0$ is rejected with $p$-value = 0.03, $\alpha = 0.05$.

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

Flashcard 32: Identify the alternative hypothesis for a one-tailed test where $p_1 < p_2$.

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

Flashcard 33: State the alternative hypothesis for testing the difference between two proportions.

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

Flashcard 34: Which test is used for large sample sizes in comparing two proportions?

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

Flashcard 35: Find the critical value for a 99% confidence interval.

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

Flashcard 36: Find the critical value for a two-tailed test at 5% significance level.

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

Flashcard 37: What is the condition for normality in a two-proportion z-test?

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

Flashcard 38: State the null hypothesis for testing the difference between two proportions.

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

Flashcard 39: Identify the formula for calculating $\hat{p}_1$ in a sample.

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

Flashcard 40: Find the critical value for a two-tailed test at 5% significance level.

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

Flashcard 41: Determine if $H_0$ is rejected with $p$-value = 0.03, $\alpha = 0.05$.

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

Flashcard 42: What is the critical value for a one-tailed test at 5% significance level?

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

Flashcard 43: Calculate $\hat{p}_2$ with $x_2 = 50$, $n_2 = 150$.

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

Flashcard 44: What is the condition for normality in a two-proportion z-test?

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

Flashcard 45: What is the assumption regarding independence in a two-proportion z-test?

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

Flashcard 46: Identify the formula for calculating $\hat{p}_2$ in a sample.

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

Flashcard 47: Find the $z$-score for $\hat{p}_1 = 0.4$, $\hat{p}_2 = 0.5$, $n_1 = n_2 = 100$.

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

Flashcard 48: What is the purpose of a two-proportion z-test?

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

Flashcard 49: What is the formula for the test statistic in a two-proportion z-test?

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

Flashcard 50: What is the assumption regarding independence in a two-proportion z-test?

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

Flashcard 51: What is the condition for sample size in a two-proportion z-test?

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

Flashcard 52: What is the condition for sample size in a two-proportion z-test?

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

Flashcard 53: Identify the significance level if $z = 1.96$ at a two-tailed test.

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

Flashcard 54: Find the $z$-score if $\hat{p}_1 = 0.6$, $\hat{p}_2 = 0.4$, $n_1 = n_2 = 50$.

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

Flashcard 55: Calculate the standard error for a difference in sample proportions.

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

Flashcard 56: Calculate $\hat{p}$ if $x_1 = 10$, $x_2 = 15$, $n_1 = 50$, $n_2 = 60$.

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

Flashcard 57: Which test is used for large sample sizes in comparing two proportions?

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

Flashcard 58: Calculate $\hat{p}_1$ with $x_1 = 40$, $n_1 = 120$.

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

Flashcard 59: Identify the alternative hypothesis for a one-tailed test where $p_1 > p_2$.

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

Flashcard 60: What is the formula for the test statistic in a two-proportion z-test?

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

Flashcard 61: Identify the alternative hypothesis for a one-tailed test where $p_1 > p_2$.

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

Flashcard 62: Identify the significance level if $z = 1.96$ at a two-tailed test.

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

Flashcard 63: State the meaning of a p-value in hypothesis testing.

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

Flashcard 64: Find the critical value for a 99% confidence interval.

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

Flashcard 65: What is the effect of increasing sample size on the standard error?

Answer: Standard error decreases. Larger samples reduce sampling variability.

Flashcard 66: What does a z-test assume about the sampling distribution?

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

Flashcard 67: Find the $z$-score for $\hat{p}_1 = 0.4$, $\hat{p}_2 = 0.5$, $n_1 = n_2 = 100$.

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

Flashcard 68: State the meaning of a p-value in hypothesis testing.

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

Flashcard 69: Identify the statistical test for $n_1 = 30$, $n_2 = 40$, $x_1 = 15$, $x_2 = 20$.

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

Flashcard 70: Identify the formula for calculating $\hat{p}_2$ in a sample.

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

Flashcard 71: Calculate $\hat{p}_2$ with $x_2 = 50$, $n_2 = 150$.

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

Flashcard 72: What is the role of the standard error in a two-proportion z-test?

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

Flashcard 73: Calculate the standard error for a difference in sample proportions.

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

Flashcard 74: Calculate the pooled proportion for $x_1 = 25$, $x_2 = 35$, $n_1 = 50$, $n_2 = 70$.

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