Difference of Two Population Proportions (Test) - AP Statistics
Card 1 of 30
What is the condition for sample size in a two-proportion z-test?
What is the condition for sample size in a two-proportion z-test?
Tap to reveal answer
Each group should have at least 10 successes and 10 failures. Ensures normal approximation is valid for both samples.
Each group should have at least 10 successes and 10 failures. Ensures normal approximation is valid for both samples.
← Didn't Know|Knew It →
Find the $z$-score if $\hat{p}_1 = 0.6$, $\hat{p}_2 = 0.4$, $n_1 = n_2 = 50$.
Find the $z$-score if $\hat{p}_1 = 0.6$, $\hat{p}_2 = 0.4$, $n_1 = n_2 = 50$.
Tap to reveal answer
$z = 2.83$. Using pooled $\hat{p} = 0.5$ in standard error calculation.
$z = 2.83$. Using pooled $\hat{p} = 0.5$ in standard error calculation.
← Didn't Know|Knew It →
What is the assumption regarding independence in a two-proportion z-test?
What is the assumption regarding independence in a two-proportion z-test?
Tap to reveal answer
Samples must be independent. Two samples must be selected independently.
Samples must be independent. Two samples must be selected independently.
← Didn't Know|Knew It →
Calculate the pooled proportion for $x_1 = 25$, $x_2 = 35$, $n_1 = 50$, $n_2 = 70$.
Calculate the pooled proportion for $x_1 = 25$, $x_2 = 35$, $n_1 = 50$, $n_2 = 70$.
Tap to reveal answer
$\hat{p} = 0.5$. Pooled proportion: $(25+35)/(50+70) = 0.5$.
$\hat{p} = 0.5$. Pooled proportion: $(25+35)/(50+70) = 0.5$.
← Didn't Know|Knew It →
What is the role of the standard error in a two-proportion z-test?
What is the role of the standard error in a two-proportion z-test?
Tap to reveal answer
Measures the variability of the sampling distribution. Quantifies uncertainty in the difference estimate.
Measures the variability of the sampling distribution. Quantifies uncertainty in the difference estimate.
← Didn't Know|Knew It →
What is the formula for calculating a confidence interval for $p_1-p_2$?
What is the formula for calculating a confidence interval for $p_1-p_2$?
Tap to reveal 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.
$(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$. Uses separate standard errors for confidence interval.
← Didn't Know|Knew It →
Find $\hat{p}$ using $x_1 = 30$, $x_2 = 40$, $n_1 = 100$, $n_2 = 100$.
Find $\hat{p}$ using $x_1 = 30$, $x_2 = 40$, $n_1 = 100$, $n_2 = 100$.
Tap to reveal answer
$\hat{p} = 0.35$. Pooled proportion: $(30+40)/(100+100) = 0.35$.
$\hat{p} = 0.35$. Pooled proportion: $(30+40)/(100+100) = 0.35$.
← Didn't Know|Knew It →
Identify the formula for calculating $\hat{p}_2$ in a sample.
Identify the formula for calculating $\hat{p}_2$ in a sample.
Tap to reveal answer
$\hat{p}_2 = \frac{x_2}{n_2}$. Second sample proportion formula for group 2.
$\hat{p}_2 = \frac{x_2}{n_2}$. Second sample proportion formula for group 2.
← Didn't Know|Knew It →
Identify the formula for calculating $\hat{p}_1$ in a sample.
Identify the formula for calculating $\hat{p}_1$ in a sample.
Tap to reveal answer
$\hat{p}_1 = \frac{x_1}{n_1}$. Sample proportion equals successes divided by sample size.
$\hat{p}_1 = \frac{x_1}{n_1}$. Sample proportion equals successes divided by sample size.
← Didn't Know|Knew It →
Calculate the standard error for a difference in sample proportions.
Calculate the standard error for a difference in sample proportions.
Tap to reveal answer
$\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}$. Measures variability of 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.
← Didn't Know|Knew It →
What is the purpose of a two-proportion z-test?
What is the purpose of a two-proportion z-test?
Tap to reveal answer
To compare two population proportions. Determines if two population proportions differ significantly.
To compare two population proportions. Determines if two population proportions differ significantly.
← Didn't Know|Knew It →
Find the critical value for a two-tailed test at 5% significance level.
Find the critical value for a two-tailed test at 5% significance level.
Tap to reveal answer
Approximately $\pm 1.96$. Standard normal distribution critical values at $\alpha = 0.05$.
Approximately $\pm 1.96$. Standard normal distribution critical values at $\alpha = 0.05$.
← Didn't Know|Knew It →
Determine the degrees of freedom used in a two-proportion z-test.
Determine the degrees of freedom used in a two-proportion z-test.
Tap to reveal answer
Not applicable; z-test uses standard normal distribution. Z-test uses normal distribution, not t-distribution.
Not applicable; z-test uses standard normal distribution. Z-test uses normal distribution, not t-distribution.
← Didn't Know|Knew It →
Identify the pooled sample proportion formula in a two-proportion test.
Identify the pooled sample proportion formula in a two-proportion test.
Tap to reveal answer
$\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$. Combines successes from both samples over total sample size.
$\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$. Combines successes from both samples over total sample size.
← Didn't Know|Knew It →
State the null hypothesis for testing the difference between two proportions.
State the null hypothesis for testing the difference between two proportions.
Tap to reveal answer
$H_0: p_1 - p_2 = 0$. Assumes no difference between population proportions.
$H_0: p_1 - p_2 = 0$. Assumes no difference between population proportions.
← Didn't Know|Knew It →
State the alternative hypothesis for testing the difference between two proportions.
State the alternative hypothesis for testing the difference between two proportions.
Tap to reveal answer
$H_a: p_1 - p_2 \neq 0$ (or $>0$ or $<0$). Tests for significant difference in either direction or one-sided.
$H_a: p_1 - p_2 \neq 0$ (or $>0$ or $<0$). Tests for significant difference in either direction or one-sided.
← Didn't Know|Knew It →
What is the formula for the test statistic in a two-proportion z-test?
What is the formula for the test statistic in a two-proportion z-test?
Tap to reveal 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.
$z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$. Uses pooled proportion for standard error under null hypothesis.
← Didn't Know|Knew It →
Calculate $\hat{p}$ if $x_1 = 10$, $x_2 = 15$, $n_1 = 50$, $n_2 = 60$.
Calculate $\hat{p}$ if $x_1 = 10$, $x_2 = 15$, $n_1 = 50$, $n_2 = 60$.
Tap to reveal answer
$\hat{p} = 0.227$. Pooled proportion: $(10+15)/(50+60) = 0.227$.
$\hat{p} = 0.227$. Pooled proportion: $(10+15)/(50+60) = 0.227$.
← Didn't Know|Knew It →
What does a two-proportion z-test evaluate?
What does a two-proportion z-test evaluate?
Tap to reveal answer
Difference between two population proportions. Tests whether two population proportions are equal.
Difference between two population proportions. Tests whether two population proportions are equal.
← Didn't Know|Knew It →
What is the condition for normality in a two-proportion z-test?
What is the condition for normality in a two-proportion z-test?
Tap to reveal answer
Sample sizes are large enough for a normal approximation. Large sample sizes ensure sampling distribution normality.
Sample sizes are large enough for a normal approximation. Large sample sizes ensure sampling distribution normality.
← Didn't Know|Knew It →
Identify the statistical test for $n_1 = 30$, $n_2 = 40$, $x_1 = 15$, $x_2 = 20$.
Identify the statistical test for $n_1 = 30$, $n_2 = 40$, $x_1 = 15$, $x_2 = 20$.
Tap to reveal answer
Two-proportion z-test. Sample sizes meet conditions for normal approximation.
Two-proportion z-test. Sample sizes meet conditions for normal approximation.
← Didn't Know|Knew It →
Determine if $H_0$ is rejected with $p$-value = 0.03, $\alpha = 0.05$.
Determine if $H_0$ is rejected with $p$-value = 0.03, $\alpha = 0.05$.
Tap to reveal answer
Yes, reject $H_0$. P-value less than significance level, reject null.
Yes, reject $H_0$. P-value less than significance level, reject null.
← Didn't Know|Knew It →
Identify the null hypothesis if $p_1 = p_2$ is assumed.
Identify the null hypothesis if $p_1 = p_2$ is assumed.
Tap to reveal answer
$H_0: p_1 - p_2 = 0$. Null hypothesis assumes equal population proportions.
$H_0: p_1 - p_2 = 0$. Null hypothesis assumes equal population proportions.
← Didn't Know|Knew It →
Find the critical value for a 99% confidence interval.
Find the critical value for a 99% confidence interval.
Tap to reveal answer
Approximately $\pm 2.576$. 99% confidence level critical values from normal distribution.
Approximately $\pm 2.576$. 99% confidence level critical values from normal distribution.
← Didn't Know|Knew It →
What is the effect of increasing sample size on the standard error?
What is the effect of increasing sample size on the standard error?
Tap to reveal answer
Standard error decreases. Larger samples reduce sampling variability.
Standard error decreases. Larger samples reduce sampling variability.
← Didn't Know|Knew It →
Find $\hat{p}$ using $x_1 = 30$, $x_2 = 40$, $n_1 = 100$, $n_2 = 100$.
Find $\hat{p}$ using $x_1 = 30$, $x_2 = 40$, $n_1 = 100$, $n_2 = 100$.
Tap to reveal answer
$\hat{p} = 0.35$. Pooled proportion: $(30+40)/(100+100) = 0.35$.
$\hat{p} = 0.35$. Pooled proportion: $(30+40)/(100+100) = 0.35$.
← Didn't Know|Knew It →
Calculate $\hat{p}$ if $x_1 = 10$, $x_2 = 15$, $n_1 = 50$, $n_2 = 60$.
Calculate $\hat{p}$ if $x_1 = 10$, $x_2 = 15$, $n_1 = 50$, $n_2 = 60$.
Tap to reveal answer
$\hat{p} = 0.227$. Pooled proportion: $(10+15)/(50+60) = 0.227$.
$\hat{p} = 0.227$. Pooled proportion: $(10+15)/(50+60) = 0.227$.
← Didn't Know|Knew It →
Identify the statistical test for $n_1 = 30$, $n_2 = 40$, $x_1 = 15$, $x_2 = 20$.
Identify the statistical test for $n_1 = 30$, $n_2 = 40$, $x_1 = 15$, $x_2 = 20$.
Tap to reveal answer
Two-proportion z-test. Sample sizes meet conditions for normal approximation.
Two-proportion z-test. Sample sizes meet conditions for normal approximation.
← Didn't Know|Knew It →
Calculate the standard error for a difference in sample proportions.
Calculate the standard error for a difference in sample proportions.
Tap to reveal answer
$\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}$. Measures variability of 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.
← Didn't Know|Knew It →
Identify the formula for calculating $\hat{p}_2$ in a sample.
Identify the formula for calculating $\hat{p}_2$ in a sample.
Tap to reveal answer
$\hat{p}_2 = \frac{x_2}{n_2}$. Second sample proportion formula for group 2.
$\hat{p}_2 = \frac{x_2}{n_2}$. Second sample proportion formula for group 2.
← Didn't Know|Knew It →