An anger-management course claims that, after completing its seminar, participants will lose their tempers less often. Always a skeptic, you decide to test this claim. A random sample of 12
 seminar participants is chosen, and these participants are asked to record the number of times that they lost their tempers in the two weeks prior to the course. After the course is over, the same participants are asked to record the number of times that they lost their tempers in the next two weeks. The following table lists the results of the survey. Using these data, test the claim at the 0.10
 level of significance assuming that the population distribution of the paired differences is approximately normal. Let d=
 (participants after completing the anger-management course) −
(participants before completing the anger-management course).
 
Number of Times Temper Was Lost during a Two-Week Period
Before	10
11
6
3
4
5
9
12
12
11
4
9
After	8
12
6
4
5
4
4
11
13
12
5
7
 
Step 2 of 3 :  Compute the value of the test statistic. Round your answer to three decimal places.

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.

An anger-management course claims that, after completing its seminar, participants will lose their tempers less often. Always a skeptic, you decide to test this claim. A random sample of 12 seminar participants is chosen, and these participants are asked to record the number of times that they lost their tempers in the two weeks prior to the course. After the course is over, the same participants are asked to record the number of times that they lost their tempers in the next two weeks. The following table lists the results of the survey. Using these data, test the claim at the 0.10 level of significance assuming that the population distribution of the paired differences is approximately normal. Let d= (participants after completing the anger-management course) − (participants before completing the anger-management course). Step 2 of 3 : Compute the value of the test statistic. Round your answer to three decimal places.

Test the claim of an anger-management course's effectiveness using a paired-sample t-test. A sample of 12 participants recorded their temper loss before and after the seminar. Learn to compute the test statistic and assess the significance at a 0.10 level, using the paired t-test formula and normal distribution assumptions.

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