To solve this hypothesis testing problem, we will follow these steps:

Problem Outline

We are testing the hypothesis:
$H_0: \mu_1 - \mu_2 = 0 \quad \text{against} \quad H_1: \mu_1 - \mu_2 \neq 0$ The data contains two samples, with standard deviations $\sigma_1 = 3.5$ and $\sigma_2 = 1.3$. The significance level is $\alpha = 0.05$. Our goal is to:

1. Calculate the test statistic $Z$.
2. Find the $P$-value.
3. Determine if $H_0$ is rejected.

Step 1: Compute the sample means ($\bar{x}_1$ and $\bar{x}_2$)

Sample 1:

The data for Sample 1 is: $36, 38, 31, 32, 30, 31, 29, 39, 31, 37, 36, 32, 39, 34, 38$

$\bar{x}_1 = \frac{\text{Sum of all Sample 1 values}}{\text{Number of Sample 1 values}}$ $\bar{x}_1 = \frac{36 + 38 + 31 + 32 + 30 + 31 + 29 + 39 + 31 + 37 + 36 + 32 + 39 + 34 + 38}{15}$ $\bar{x}_1 = \frac{513}{15} = 34.2$

Sample 2:

The data for Sample 2 is: $33, 29, 32, 30, 36, 31, 33, 30, 31, 30, 31, 32, 34, 32, 34$

$\bar{x}_2 = \frac{\text{Sum of all Sample 2 values}}{\text{Number of Sample 2 values}}$ $\bar{x}_2 = \frac{33 + 29 + 32 + 30 + 36 + 31 + 33 + 30 + 31 + 30 + 31 + 32 + 34 + 32 + 34}{15}$ $\bar{x}_2 = \frac{448}{15} = 29.87$

Step 2: Calculate the test statistic $Z$

The test statistic is given by: $Z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$

Where:

• $\bar{x}_1 = 34.2$
• $\bar{x}_2 = 29.87$
• $\sigma_1 = 3.5$, $n_1 = 15$
• $\sigma_2 = 1.3$, $n_2 = 15$

Substitute the values: $Z = \frac{34.2 - 29.87}{\sqrt{\frac{3.5^2}{15} + \frac{1.3^2}{15}}}$ $Z = \frac{4.33}{\sqrt{\frac{12.25}{15} + \frac{1.69}{15}}}$ $Z = \frac{4.33}{\sqrt{0.8167 + 0.1127}}$ $Z = \frac{4.33}{\sqrt{0.9294}}$ $Z = \frac{4.33}{0.9641} = 4.49$

Step 3: Find the $P$-value

Using the standard normal distribution table (or a calculator): $P(Z > 4.49) \approx 0$

Since this is a two-tailed test, the $P$-value is: $P = 2 \times P(Z > 4.49) \approx 0$

Step 4: Decision

The $P$-value ($0$) is less than the significance level ($\alpha = 0.05$). Therefore, we reject $H_0$ and conclude that there is a significant difference between $\mu_1$ and $\mu_2$.

Final Results:

1. Test Statistic: $Z = 4.49$
2. $P$-value: $0$
3. Decision: Reject $H_0$

Would you like me to provide further clarifications or verify anything else?

Additional Questions:

1. How is the test statistic formula derived for two independent samples?
2. What is the interpretation of rejecting the null hypothesis in practical terms?
3. How does the choice of $\alpha$ affect hypothesis testing?
4. Could this problem be solved using a confidence interval? How?
5. What assumptions are required for using the $Z$-test in this scenario?

Tip:

Always double-check your calculations for means and standard deviations, as small errors can propagate through to the final test statistic!