H 0 : μ 2 − μ 1

Let's analyze the hypothesis test step by step.

1. Type of Test

The hypotheses are given as:

• Null Hypothesis ($H_0$): $\mu_2 - \mu_1 = 0$
• Alternative Hypothesis ($H_1$): $\mu_2 - \mu_1 \neq 0$

Since the alternative hypothesis is looking for any difference (either positive or negative) between the two means, this is a two-tailed test.

2. Critical Value

For a two-tailed test with a significance level $\alpha = 0.01$, the critical values correspond to the area in both tails of the normal distribution. The critical value for a two-tailed test at $\alpha = 0.01$ can be found using the standard normal distribution (Z-distribution) table.

At $\alpha = 0.01$, the area in each tail is $\alpha/2 = 0.005$. The Z-score corresponding to 0.005 in the tail is approximately 2.576.

So, the positive critical value is: $\text{Critical value} = 2.576$

3. Test Statistic

To calculate the test statistic, we use the formula for the two-sample Z-test for means:

$Z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

Given: $\bar{x}_1 = 50.4, \quad s_1 = 2.14, \quad n_1 = 11$ $\bar{x}_2 = 47.5, \quad s_2 = 2.36, \quad n_2 = 13$ $\mu_1 - \mu_2 = 0$

Substituting these values:

$Z = \frac{(50.4 - 47.5) - 0}{\sqrt{\frac{2.14^2}{11} + \frac{2.36^2}{13}}}$

Now, calculate the value step by step:

$Z = \frac{2.9}{\sqrt{\frac{4.5796}{11} + \frac{5.5696}{13}}}$ $Z = \frac{2.9}{\sqrt{0.416327 + 0.4284308}} = \frac{2.9}{\sqrt{0.8447578}} = \frac{2.9}{0.919144} \approx 3.155$

So, the test statistic $Z$ is approximately 3.155.

4. Decision

The decision rule is:

• Reject the null hypothesis if $|Z| > 2.576$ (since this is a two-tailed test).

Since the calculated test statistic $Z = 3.155$ is greater than the critical value $2.576$, we reject the null hypothesis.

Summary:

• Type of test: Two-tailed test
• Critical value: 2.576
• Test statistic: 3.155
• Decision: Reject the null hypothesis

Would you like further details or have any questions about the steps involved?

Here are 8 related questions:

1. What is the difference between a one-tailed and a two-tailed test?
2. How is the critical value determined for different significance levels?
3. What assumptions are required for the two-sample Z-test to be valid?
4. How would the test statistic change if the sample sizes were larger?
5. What is the relationship between the p-value and the test statistic?
6. How does the choice of $\alpha$ affect the outcome of the hypothesis test?
7. What are Type I and Type II errors in the context of hypothesis testing?
8. How would you interpret the test results if the null hypothesis was not rejected?

Tip: Always check if the conditions for the hypothesis test are met, such as normality of data and equal variances, before proceeding with the test.