The document provides various hypothesis testing problems involving population proportions, means, and variances using different significance levels. Here’s a summary and guidance for solving these problems:

1. Two-Proportion Z-Test: Problems involve comparing proportions from two populations using the normal approximation. Key steps:

   • Null hypothesis ($H_0$): Usually $p_1 = p_2$.
   • Calculate the test statistic using the formula: $z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$ where $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$.
   • Compare the p-value against the significance level $\alpha$ to make a decision.

2. Two-Sample T-Test: Problems involve comparing means of two independent populations. The approach differs depending on whether variances are assumed equal or unequal:

   • Null hypothesis ($H_0$): Usually $\mu_1 = \mu_2$.
   • Use the t-statistic formula: $t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$ If variances are unequal, Welch's t-test is applied.
   • Compare the p-value against $\alpha$.

3. Tests for Right-Tailed, Left-Tailed, and Two-Tailed Hypotheses:

   • Right-tailed: Testing if one population parameter is greater than the other.
   • Left-tailed: Testing if one population parameter is less than the other.
   • Two-tailed: Testing for any difference between population parameters.

4. Steps for Conclusion:

   • Calculate the test statistic (z or t).
   • Determine the p-value using the appropriate distribution table.
   • Compare the p-value to $\alpha$ (significance level).
   • Make a decision: Reject $H_0$ if p-value $\leq \alpha$, otherwise fail to reject $H_0$.

If you need detailed solutions for a specific question, let me know which one to start with! Here are 5 questions you can consider:

1. How is the test statistic calculated for two-proportion tests?
2. What is the difference between a two-tailed and a one-tailed test?
3. How do you determine whether variances are equal or unequal in two-sample t-tests?
4. How can we interpret the p-value in hypothesis testing?
5. What adjustments are needed if sample sizes are small?

Tip: Always double-check assumptions for normality or variance equality before applying a specific test.