To address the problem, we'll follow the standard procedure for hypothesis testing. Let's go through each part step by step.

Part 1: State the Hypotheses

The null hypothesis (H₀) and the alternative hypothesis (H₁) are given as:

• Null Hypothesis (H₀): $\mu = 25$
• Alternative Hypothesis (H₁): $\mu \neq 25$

Part 2: Calculate the P-value

We need to perform a t-test to determine if the sample mean is significantly different from the population mean. The formula for the test statistic (t) is:

$t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$

Where:

• $\bar{x} = 22.2$ (sample mean)
• $\mu = 25$ (population mean)
• $s = 6.3$ (sample standard deviation)
• $n = 15$ (sample size)

First, we calculate the test statistic $t$:

$t = \frac{22.2 - 25}{6.3 / \sqrt{15}}$

Calculate the denominator:

$\frac{6.3}{\sqrt{15}} = \frac{6.3}{3.87298} \approx 1.6266$

Then the test statistic:

$t = \frac{22.2 - 25}{1.6266} \approx \frac{-2.8}{1.6266} \approx -1.721$

Next, we find the degrees of freedom (df):

$df = n - 1 = 15 - 1 = 14$

Using a t-distribution table or a calculator for a two-tailed test, we find the p-value for $t = -1.721$ with 14 degrees of freedom.

According to the information given in the question, the p-value is approximately 0.107.

Part 3: State the Conclusion

We compare the p-value with the significance level ($\alpha$) to make a decision. Given:

• $\alpha = 0.01$
• $\text{p-value} = 0.107$

Since the p-value (0.107) is greater than the significance level (0.01), we do not reject the null hypothesis.

Therefore, the correct conclusion is:

C. Do not reject $H_0$ because the p-value is greater than the $\alpha = 0.01$ level of significance.

Summary

• Hypotheses: $H_0: \mu = 25$, $H_1: \mu \neq 25$
• P-value: 0.107
• Conclusion: Do not reject $H_0$

Would you like more details or have any questions?

Here are five questions you can ask next:

1. How is the t-distribution different from the normal distribution?
2. What are the assumptions of the t-test?
3. How can we interpret the p-value in hypothesis testing?
4. Can we use this method if the population is not normally distributed?
5. How does sample size affect the t-test?

Tip: When conducting hypothesis tests, always ensure your sample size and data meet the necessary assumptions to validate the results.