What is the Two-Sample Z-Test for Population Proportions?

📚 Quick Study Guide

• 🎯 Purpose: To determine if there is a significant difference between the proportions of two independent populations.
• 📝 Null Hypothesis ($H_0$): The two population proportions are equal ($p_1 = p_2$).
• 🧪 Alternative Hypothesis ($H_1$): The two population proportions are not equal ($p_1 \neq p_2$), or $p_1 > p_2$, or $p_1 < p_2$.
• 🔢 Test Statistic (Z): $Z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$, where $\hat{p}_1$ and $\hat{p}_2$ are the sample proportions, $n_1$ and $n_2$ are the sample sizes, and $\hat{p}$ is the pooled sample proportion.
• 📊 Pooled Sample Proportion ($\hat{p}$): $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$, where $x_1$ and $x_2$ are the number of successes in each sample.
• 📈 Decision Rule: If the absolute value of the calculated Z-statistic is greater than the critical Z-value (based on the chosen significance level $\alpha$), reject the null hypothesis.
• 💡 Assumptions: Independent samples, simple random samples, and $n_1p_1$, $n_1(1-p_1)$, $n_2p_2$, and $n_2(1-p_2)$ are all greater than or equal to 10.

Practice Quiz

1. Question 1: What is the primary purpose of the Two-Sample Z-Test for Population Proportions?
   1. A. To compare the means of two populations.
   2. B. To compare the proportions of two populations.
   3. C. To determine the correlation between two variables.
   4. D. To analyze the variance within a single population.
2. Question 2: The null hypothesis ($H_0$) for the Two-Sample Z-Test for Population Proportions typically states that:
   1. A. The two population proportions are not equal.
   2. B. The two population proportions are equal.
   3. C. The population proportion is greater than a certain value.
   4. D. The population proportion is less than a certain value.
3. Question 3: What does $\hat{p}$ represent in the Z-test statistic formula?
   1. A. The sample proportion of the first population.
   2. B. The sample proportion of the second population.
   3. C. The pooled sample proportion.
   4. D. The population proportion.
4. Question 4: In the formula $Z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$, what do $n_1$ and $n_2$ represent?
   1. A. The sample proportions.
   2. B. The number of successes in each sample.
   3. C. The sample sizes.
   4. D. The population sizes.
5. Question 5: If the calculated Z-statistic is 2.5 and the critical Z-value at $\alpha = 0.05$ is 1.96, what is the correct decision?
   1. A. Accept the null hypothesis.
   2. B. Reject the null hypothesis.
   3. C. Fail to reject the null hypothesis.
   4. D. Increase the sample size.
6. Question 6: Which of the following is NOT a necessary assumption for the Two-Sample Z-Test for Population Proportions?
   1. A. Independent samples.
   2. B. Simple random samples.
   3. C. Equal sample sizes.
   4. D. $n_1p_1$, $n_1(1-p_1)$, $n_2p_2$, and $n_2(1-p_2)$ are all greater than or equal to 10.
7. Question 7: The pooled sample proportion $\hat{p}$ is calculated as:
   1. A. $\frac{n_1 + n_2}{x_1 + x_2}$
   2. B. $\frac{x_1 + x_2}{n_1 + n_2}$
   3. C. $\frac{x_1}{n_1} + \frac{x_2}{n_2}$
   4. D. $\frac{n_1}{x_1} + \frac{n_2}{x_2}$