Online Quiz: Master the Two-Sample Z-Test for Population Proportions

📚 Quick Study Guide

• 🎯 Purpose: The two-sample z-test for population proportions determines if there's a statistically significant difference between the proportions of two independent groups.
• 🔑 Assumptions:
  • 🌱 Both samples are randomly selected and independent.
  • ✨ Each sample should be large enough to satisfy the conditions $np \geq 10$ and $n(1-p) \geq 10$ for both populations, where $n$ is the sample size and $p$ is the sample proportion.
• ➗ Pooled Proportion:
  • 🧮 Calculate the pooled sample proportion: $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$, where $x_1$ and $x_2$ are the number of successes in each sample, and $n_1$ and $n_2$ are the sample sizes.
• 📐 Test Statistic:
  • 📊 Compute the z-test statistic: $z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$, where $\hat{p}_1$ and $\hat{p}_2$ are the sample proportions for each group.
• ⚖️ Hypotheses:
  • 📝 Null Hypothesis ($H_0$): $p_1 = p_2$ (There is no difference between the population proportions).
  • 🤔 Alternative Hypothesis ($H_1$):
    • $p_1 \neq p_2$ (Two-tailed test)
    • $p_1 > p_2$ (Right-tailed test)
    • $p_1 < p_2$ (Left-tailed test)
• ✅ Decision Rule: Compare the calculated z-value to the critical z-value from the standard normal distribution or compute the p-value. Reject the null hypothesis if the p-value is less than the significance level ($\alpha$).

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 independent samples.
   2. B. To compare the proportions of two independent samples.
   3. C. To determine the correlation between two variables.
   4. D. To analyze the variance within a single sample.
2. Question 2: 🧪 What assumption must be met regarding sample size for a valid two-sample z-test for population proportions?
   1. A. $n \geq 30$ for both samples.
   2. B. $np \geq 5$ and $n(1-p) \geq 5$ for both samples.
   3. C. $np \geq 10$ and $n(1-p) \geq 10$ for both samples.
   4. D. The samples must be of equal size.
3. Question 3: 🔢 What is the formula for the pooled proportion ($\hat{p}$)?
   1. A. $\hat{p} = \frac{n_1 + n_2}{x_1 + x_2}$
   2. B. $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$
   3. C. $\hat{p} = \frac{x_1 - x_2}{n_1 - n_2}$
   4. D. $\hat{p} = \frac{n_1x_1 + n_2x_2}{n_1 + n_2}$
4. Question 4: 📊 What is the null hypothesis ($H_0$) in a two-sample z-test for population proportions?
   1. A. $p_1 \neq p_2$
   2. B. $p_1 > p_2$
   3. C. $p_1 < p_2$
   4. D. $p_1 = p_2$
5. Question 5: 📈 If the calculated z-value is 2.5 and the critical z-value at $\alpha = 0.05$ is 1.96 for a one-tailed test, what is the decision?
   1. A. Fail to reject the null hypothesis.
   2. B. Reject the null hypothesis.
   3. C. Increase the sample size.
   4. D. Decrease the significance level.
6. Question 6: 📉 In a study, sample 1 has 60 successes out of 200, and sample 2 has 80 successes out of 250. What are the sample proportions $\hat{p}_1$ and $\hat{p}_2$, respectively?
   1. A. 0.30 and 0.32
   2. B. 0.32 and 0.30
   3. C. 0.20 and 0.25
   4. D. 0.25 and 0.20
7. Question 7: 💡 What does a statistically significant result in a two-sample z-test for population proportions indicate?
   1. A. There is no difference between the two population proportions.
   2. B. There is a practical but not statistically significant difference.
   3. C. There is a statistically significant difference between the two population proportions.
   4. D. The sample sizes are too small to draw any conclusions.