Let's solve this step by step:

(a) State the null hypothesis $H_0$ and the alternative hypothesis $H_1$:

We are testing whether the mean test score for Version 1 ($\mu_1$) is less than the mean test score for Version 2 ($\mu_2$).

• $H_0$: $\mu_1 \geq \mu_2$
• $H_1$: $\mu_1 < \mu_2$

(b) Determine the type of test statistic to use:

Since the population standard deviations are not given but the sample standard deviations are provided, and the sample sizes are reasonably large, we use a two-sample t-test.

(c) Find the value of the test statistic:

The formula for the test statistic in a two-sample t-test is:

$t = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

Where:

• $\bar{x}_1 = 112.1$, $\bar{x}_2 = 117.9$
• $s_1 = 16.1$, $s_2 = 18.7$
• $n_1 = 75$, $n_2 = 95$

Step-by-step calculation:

1. Compute the difference in sample means: $\bar{x}_1 - \bar{x}_2 = 112.1 - 117.9 = -5.8$

2. Compute the variances divided by sample sizes: $\frac{s_1^2}{n_1} = \frac{16.1^2}{75} = \frac{259.21}{75} \approx 3.4561$ $\frac{s_2^2}{n_2} = \frac{18.7^2}{95} = \frac{349.69}{95} \approx 3.6809$

3. Add these variances: $\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = \sqrt{3.4561 + 3.6809} = \sqrt{7.137} \approx 2.672$

4. Calculate the test statistic: $t = \frac{-5.8}{2.672} \approx -2.171$

So, the test statistic is approximately $t = -2.171$.

(d) Find the critical value at the 0.01 level of significance:

For a one-tailed t-test at $\alpha = 0.01$ with degrees of freedom approximated using the smaller of $n_1 - 1$ or $n_2 - 1$ ($df \approx 74$), the critical value from the t-distribution table is approximately:

$t_{critical} = -2.385$

(e) Can we support the claim?

Compare $t$ with $t_{critical}$:

• Since $t = -2.171$ is not less than $t_{critical} = -2.385$, we fail to reject the null hypothesis.

Thus, we cannot support the claim that the mean test score for Version 1 is less than the mean test score for Version 2 at the 0.01 level of significance.

Do you have any questions or need further clarifications? Here are five related questions for practice:

1. How does the level of significance ($\alpha$) affect the hypothesis test conclusion?
2. What is the importance of sample size in hypothesis testing?
3. What would happen if we used a two-tailed test instead of a one-tailed test in this scenario?
4. How would changing the sample means or standard deviations impact the test statistic?
5. Why is it necessary to assume the population standard deviations are approximately equal?

Tip: Always carefully determine whether a one-tailed or two-tailed test is required based on the research hypothesis.