To calculate the p-value for this test, we will follow these steps:

Step 1: Define Hypotheses

• Null Hypothesis (H₀): $\mu_1 = \mu_2$
• Alternative Hypothesis (Hₐ): $\mu_1 > \mu_2$

This is a one-tailed test since the alternative hypothesis suggests that $\mu_1$ is greater than $\mu_2$.

Step 2: Check Assumptions

• Populations are normally distributed.
• Population variances are assumed to be equal but unknown.

Step 3: Pooled Standard Deviation

Since the population variances are assumed to be equal, we calculate the pooled standard deviation $S_p$:

$S_p = \sqrt{\frac{(n_1 - 1) \cdot SD_1^2 + (n_2 - 1) \cdot SD_2^2}{n_1 + n_2 - 2}}$

Given:

• $n_1 = 12$, $SD_1 = 12.6$
• $n_2 = 18$, $SD_2 = 14.2$

$S_p = \sqrt{\frac{(12 - 1) \cdot 12.6^2 + (18 - 1) \cdot 14.2^2}{12 + 18 - 2}}$ $S_p = \sqrt{\frac{11 \cdot 158.76 + 17 \cdot 201.64}{28}}$ $S_p = \sqrt{\frac{1746.36 + 3427.88}{28}} = \sqrt{\frac{5174.24}{28}} = \sqrt{184.08} \approx 13.57$

Step 4: Test Statistic (t-value)

The test statistic for a two-sample t-test is:

$t = \frac{M_1 - M_2}{S_p \cdot \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$

Substitute the values:

$t = \frac{56.6 - 52.6}{13.57 \cdot \sqrt{\frac{1}{12} + \frac{1}{18}}}$ $t = \frac{4}{13.57 \cdot \sqrt{0.0833 + 0.0556}}$ $t = \frac{4}{13.57 \cdot \sqrt{0.1389}} = \frac{4}{13.57 \cdot 0.3726} \approx \frac{4}{5.055} \approx 0.791$

Step 5: Degrees of Freedom (df)

To be conservative, we use the smaller sample size - 1 to estimate the degrees of freedom (df):

$df = n_1 - 1 = 12 - 1 = 11$

Step 6: p-value

Using a t-distribution table or software, look up the p-value for $t = 0.791$ with 11 degrees of freedom in a one-tailed test at $\alpha = 0.01$.

For $t = 0.791$ and $df = 11$, the corresponding p-value is approximately 0.2197.

Conclusion:

Since the p-value of 0.2197 is greater than the significance level $\alpha = 0.01$, we fail to reject the null hypothesis. There is not enough evidence to support that $\mu_1 > \mu_2$.

$\text{p-value} \approx 0.2197$

Let me know if you want more details or have questions!

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Here are 5 related questions:

1. How do you interpret a p-value in the context of hypothesis testing?
2. What assumptions are necessary for conducting a t-test?
3. What is the difference between one-tailed and two-tailed tests?
4. How does sample size affect the test statistic and p-value?
5. What is the impact of unequal variances on this test?

Tip: Always double-check if your hypothesis test should be one-tailed or two-tailed, as this affects how the p-value is interpreted.