Hypothesis Testing Steps for the Two-Sample Z-Test for Proportions

📚 Quick Study Guide

• 📊 Purpose: The two-sample z-test for proportions is used to determine if there is a significant difference between the proportions of two independent groups.
• 📝 Hypotheses:
  • Null Hypothesis ($H_0$): $p_1 = p_2$ (The proportions are equal)
  • Alternative Hypothesis ($H_1$):
    • $p_1 \neq p_2$ (Two-tailed test: The proportions are not equal)
    • $p_1 > p_2$ (Right-tailed test: Proportion 1 is greater than proportion 2)
    • $p_1 < p_2$ (Left-tailed test: Proportion 1 is less than proportion 2)
• 🔢 Test Statistic: The z-test statistic is calculated as: $z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$ 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 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.
• ✅ Steps:
  1. State the null and alternative hypotheses.
  2. Determine the significance level ($\alpha$).
  3. Calculate the test statistic.
  4. Determine the p-value.
  5. Make a decision: If p-value $\leq \alpha$, reject $H_0$.
  6. Interpret the results.
• 🔑 Assumptions:
  • The two samples are independent.
  • Both samples are simple random samples.
  • $n_1p_1$, $n_1(1-p_1)$, $n_2p_2$, and $n_2(1-p_2)$ are all greater than or equal to 10 (or 5, depending on the source) to ensure normality.

🧪 Practice Quiz

1. Question 1: What is the null hypothesis for a two-sample z-test for proportions?
   1. A) $p_1 \neq p_2$
   2. B) $p_1 > p_2$
   3. C) $p_1 = p_2$
   4. D) $p_1 < p_2$
2. Question 2: What does a two-tailed alternative hypothesis indicate in a two-sample z-test for proportions?
   1. A) $p_1$ is greater than $p_2$
   2. B) $p_1$ is less than $p_2$
   3. C) $p_1$ is not equal to $p_2$
   4. D) $p_1$ is equal to $p_2$
3. Question 3: Which of the following is the correct formula for the pooled proportion $\hat{p}$?
   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) $\frac{n_1}{x_1} + \frac{n_2}{x_2}$
4. Question 4: What is the purpose of calculating the pooled proportion in the two-sample z-test for proportions?
   1. A) To estimate the individual population proportions.
   2. B) To get a weighted average of the two sample sizes.
   3. C) To estimate the common proportion when the null hypothesis is true.
   4. D) To compare the sample sizes.
5. Question 5: What is one of the assumptions for conducting a two-sample z-test for proportions?
   1. A) The samples are dependent.
   2. B) The population standard deviations are known.
   3. C) $n_1p_1, n_1(1-p_1), n_2p_2,$ and $n_2(1-p_2)$ are all greater than or equal to 10.
   4. D) The populations are normally distributed.
6. Question 6: In hypothesis testing, if the p-value is less than or equal to the significance level ($\alpha$), what decision should be made?
   1. A) Accept the null hypothesis.
   2. B) Fail to reject the null hypothesis.
   3. C) Reject the alternative hypothesis.
   4. D) Reject the null hypothesis.
7. Question 7: A researcher wants to determine if the proportion of men who prefer Brand A is different from the proportion of women who prefer Brand A. What type of test should they use?
   1. A) One-sample t-test
   2. B) Two-sample z-test for means
   3. C) Two-sample z-test for proportions
   4. D) Paired t-test