Let's go through the questions step-by-step based on the images you provided.

Analysis of Each Part

Part (a): Setting Up the Null and Alternative Hypotheses

1. Given Information: The claim is that men in the U.S. have become taller than in previous generations.

2. Appropriate Null and Alternative Hypotheses:

   • Null Hypothesis ($H_0$): The mean height of men has not increased from previous generations, so $\mu = 178.4$.
   • Alternative Hypothesis ($H_1$): The mean height of men has increased, so $\mu > 178.4$.

   Based on the options, the correct choice is:

   • $H_0: \mu = 178.4$
   • $H_1: \mu > 178.4$

   This would be a right-tailed test since we are testing if the current mean is greater than the previous mean.

Part (b): Conditions for 1-Sample t-Test

To determine if a 1-sample t-test is appropriate, the following conditions must be checked:

1. Random Sampling: The sample of 25 men should ideally be randomly selected.
2. Independence: Each height measurement should be independent of others.
3. Normality: For a sample size of 25, if the population is approximately normal or the sample size is reasonably large (usually n ≥ 30 is standard), then the t-test can be used.

Based on the information given, it’s assumed the population is approximately normal, so the 1-sample t-test conditions appear to be satisfied.

Part (c): Sketching the Distribution and Shading the Critical Region

In a right-tailed t-test with a significance level of 5%, the critical region would be in the upper tail of the t-distribution. The t-value corresponding to a 5% significance level in the right tail would need to be identified to shade the critical region.

Part (d): Using the P-Value Approach

• P-Value Calculation: To compute the p-value, you would need the test statistic $t$ based on the sample data. This can be calculated with: $t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$ where:
  • $\bar{x}$ is the sample mean (178.4 cm),
  • $\mu$ is the hypothesized mean (e.g., 172.2 cm if comparing against previous generations),
  • $s$ is the standard deviation (7.49 cm),
  • $n$ is the sample size (25).

Once the t-value is calculated, use it to find the p-value, comparing it against 0.05.

• Comparing the P-Value: Based on whether the calculated p-value is less than 0.05, you would select either "greater" or "less."

Part (e): Decision on the Null Hypothesis

• If the p-value is less than 0.05, reject the null hypothesis, indicating that there is significant evidence to support the claim that men in the U.S. have become taller than previous generations.
• If the p-value is greater than 0.05, fail to reject the null hypothesis, indicating insufficient evidence to support the claim.

Would you like me to calculate the t-value and the p-value for you, or do you have any specific questions on any part?

Expanding Questions for Better Understanding

1. How do we interpret the p-value in hypothesis testing?
2. What is the importance of the significance level in deciding to reject the null hypothesis?
3. Why is a right-tailed test appropriate for this scenario?
4. How does the choice of sample size affect the accuracy of a 1-sample t-test?
5. What would change in the approach if the sample size were larger than 30?

Tip:

Always check that the assumptions of normality and independence are reasonably satisfied before conducting a t-test, as these assumptions help ensure the validity of the test results.