Let us address each part of this hypothesis testing problem step by step.

Part (a): Null and Alternative Hypotheses

The null and alternative hypotheses are:

$H_0: \mu = 25 \quad \text{(The population mean is equal to 25.)}$ $H_1: \mu \neq 25 \quad \text{(The population mean is not equal to 25.)}$

Part (b): Calculate the $P$-value

The test statistic for a one-sample $t$-test is calculated as:

$t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$

Substitute the given values:

• Sample mean ($\bar{x}$) = 19.1
• Hypothesized population mean ($\mu_0$) = 25
• Sample standard deviation ($s$) = 6.3
• Sample size ($n$) = 15

$t = \frac{19.1 - 25}{\frac{6.3}{\sqrt{15}}} = \frac{-5.9}{\frac{6.3}{3.873}} = \frac{-5.9}{1.625} \approx -3.63$

Using the $t$-distribution with $df = n - 1 = 14$, we calculate the two-tailed $P$-value. From $t = -3.63$, we find:

$P\text{-value} \approx 0.003$

Part (c): State the conclusion for the test

Compare the $P$-value with the significance level $\alpha = 0.01$:

$P\text{-value} = 0.003 < 0.01$

Since the $P$-value is less than $\alpha$, we reject the null hypothesis $H_0$.

Correct conclusion: D. Reject $H_0$ because the $P$-value is less than $\alpha = 0.01$.

Part (d): State the conclusion in context of the problem

Since we rejected $H_0$, we conclude:

There is sufficient evidence at the $\alpha = 0.01$ level of significance to conclude that the population mean is different from 25.

Summary:

1. Hypotheses: $H_0: \mu = 25$, $H_1: \mu \neq 25$.
2. $P$-value = 0.003.
3. Decision: Reject $H_0$.
4. Conclusion: The population mean is significantly different from 25.

Related Questions:

1. How is the $t$-statistic formula derived, and why is it used instead of $z$-statistics?
2. What assumptions must be met to perform this one-sample $t$-test?
3. How does changing $\alpha$ from 0.01 to 0.05 affect the conclusion?
4. Why is the two-tailed $P$-value used for this hypothesis test?
5. What would happen if the sample size were increased to 30? How would it impact the $P$-value?

Tip: Always check the normality assumption or use larger sample sizes to ensure the validity of $t$-tests.