This problem is asking us to conduct a paired sample hypothesis test to compare the weights of male college students in September and April using a significance level of $0.01$. Let's break it down step by step:

Part (a): Null and Alternative Hypotheses

The hypothesis test is for paired samples, where the differences $d$ are defined as:

$d = \text{April weight} - \text{September weight}$

• The null hypothesis ($H_0$) assumes that there is no difference in the mean weights between April and September: $H_0: \mu_d = 0$

• The alternative hypothesis ($H_1$) claims that the weights in April are greater than the weights in September. Thus: $H_1: \mu_d > 0$

Next Steps

• Compute the differences $d$ for each pair.

• Calculate the mean ($\bar{d}$) and standard deviation ($s_d$) of the differences.

• Perform a one-sample t-test for the differences: $t = \frac{\bar{d} - 0}{s_d / \sqrt{n}}$ where $n$ is the number of paired data points.

• Compare the calculated $t$-value with the critical $t$-value (or use the p-value) at the 0.01 significance level to make a decision.

Would you like me to walk through the full calculation with the data provided? Let me know if you have any questions!

Related Questions

1. What is the formula for calculating the standard deviation of the differences in paired data?
2. How do you determine whether the null hypothesis should be rejected or not based on the $t$-value and significance level?
3. What are the assumptions for conducting a paired $t$-test?
4. How would the hypotheses change if the claim were that weights in September were greater than in April?
5. How does the sample size $n$ affect the test's power and the reliability of the results?

Tip

Always verify that the data meets the assumptions for a paired $t$-test, including approximate normality of the differences and proper pairing of the data points!