This is a paired-sample t-test scenario, where we compare the participants' temper loss counts before and after completing the anger-management course. The differences between the paired observations (before and after) will be analyzed to determine if there is a significant change.

Let's follow these steps to compute the test statistic:

1. Define the hypothesis:

   • Null hypothesis $H_0$: $\mu_d = 0$ (There is no difference in the number of times participants lost their temper before and after the course).
   • Alternative hypothesis $H_1$: $\mu_d \neq 0$ (There is a difference in the number of times participants lost their temper before and after the course).

2. Compute the differences $d$ for each pair (After - Before):

   • For each participant, subtract the "before" value from the "after" value to get the difference.

3. Calculate the mean and standard deviation of the differences:

   • The mean of the differences $\bar{d}$ and the standard deviation of the differences $s_d$ will be used in the formula for the t-test statistic.

4. Compute the t-statistic: The formula for the paired-sample t-test is:

   $t = \frac{\bar{d}}{s_d / \sqrt{n}}$

   Where:

   • $\bar{d}$ is the mean of the differences
   • $s_d$ is the standard deviation of the differences
   • $n$ is the number of paired observations (12 in this case)

I'll now calculate these values and the test statistic.

The value of the test statistic is approximately $t = -0.767$.

Would you like more details on the process, or do you have any questions?

Here are 5 related questions you might find useful:

1. How do we interpret the p-value in hypothesis testing?
2. What are Type I and Type II errors in the context of this test?
3. How does sample size impact the t-test results?
4. What assumptions must be met for the paired-sample t-test?
5. What is the difference between a paired t-test and an independent t-test?

Tip: When conducting hypothesis tests, always check if the test assumptions (like normality for small samples) are met.