Understanding the Two-Sample Z-Test Statistic and Formula

📚 Quick Study Guide

• 🔢 Purpose: Used to determine if there is a statistically significant difference between the means of two independent groups.
• 📊 Assumptions: Data is normally distributed (or sample sizes are large enough for the Central Limit Theorem to apply), variances are known, and samples are independent.
• 📝 Null Hypothesis ($H_0$): There is no difference between the means of the two populations ($\mu_1 = \mu_2$).
• 🧪 Alternative Hypothesis ($H_1$): There is a difference between the means of the two populations ($\mu_1 \neq \mu_2$, $\mu_1 > \mu_2$, or $\mu_1 < \mu_2$).
• ➗ Test Statistic Formula: $z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$
  • $\bar{x}_1$ and $\bar{x}_2$ are the sample means.
  • $\mu_1$ and $\mu_2$ are the population means (often, their difference is 0 under $H_0$).
  • $\sigma_1^2$ and $\sigma_2^2$ are the population variances.
  • $n_1$ and $n_2$ are the sample sizes.
• 📈 Interpretation: A large absolute value of the z-statistic indicates strong evidence against the null hypothesis. Compare the calculated z-statistic to the critical value from the standard normal distribution (or calculate the p-value) to make a decision.

Practice Quiz

1. What is the primary purpose of the two-sample z-test?
1. A. To estimate population variances.
2. B. To compare the means of two independent groups.
3. C. To test for correlation between two variables.
4. D. To determine sample sizes.
2. Which of the following is an assumption of the two-sample z-test?
1. A. Data must be non-normally distributed.
2. B. Variances must be unknown.
3. C. Samples must be dependent.
4. D. Data is normally distributed (or sample sizes are large enough for the Central Limit Theorem to apply).
3. In the null hypothesis of a two-sample z-test, what is assumed about the population means?
1. A. They are unequal.
2. B. They are equal.
3. C. Their relationship is unknown.
4. D. One is always greater than the other.
4. What does a large absolute value of the z-statistic suggest?
1. A. Strong evidence in favor of the null hypothesis.
2. B. Strong evidence against the null hypothesis.
3. C. The sample sizes are too small.
4. D. The population variances are equal.
5. Given two samples with means $\bar{x}_1 = 50$ and $\bar{x}_2 = 55$, population variances $\sigma_1^2 = 100$ and $\sigma_2^2 = 121$, and sample sizes $n_1 = 25$ and $n_2 = 36$, what is the numerator of the z-test statistic (assuming the null hypothesis is true)?
1. A. 0
2. B. 5
3. C. -5
4. D. 10
6. Using the information from the previous question, which values are found in the denominator of the z-test statistic: $z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$?
1. A. $\sqrt{\frac{100}{25} + \frac{121}{36}}$
2. B. $\frac{100}{25} + \frac{121}{36}$
3. C. $\sqrt{\frac{25}{100} + \frac{36}{121}}$
4. D. $\frac{25}{100} + \frac{36}{121}$
7. If the calculated z-statistic is 2.5 and the critical value for a one-tailed test at $\alpha = 0.05$ is 1.645, what decision should be made regarding the null hypothesis?
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.