Questions: 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 0.05 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: 6, 5, 11, 7, 3, 5, 4, 9, 3, 6, 5, 5 After: 5, 4, 9, 3, 4, 6, 4, 7, 4, 7, 5, 4

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 0.05 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: 6, 5, 11, 7, 3, 5, 4, 9, 3, 6, 5, 5 After: 5, 4, 9, 3, 4, 6, 4, 7, 4, 7, 5, 4
Transcript text: 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 0.05 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). \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \multicolumn{10}{|c|}{ Number of Times Temper Was Lost during a Two-Week Period } \\ \hline Before & 6 & 5 & 11 & 7 & 3 & 5 & 4 & 9 & 3 & 6 & 5 & 5 \\ \hline After & 5 & 4 & 9 & 3 & 4 & 6 & 4 & 7 & 4 & 7 & 5 & 4 \\ \hline \end{tabular} Copy Data Step 2 of 3: Compute the value of the test statistic. Round your answer to three decimal places.
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Solution

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Solution Steps

To test the claim that participants lose their tempers less often after completing the anger-management course, we can perform a paired t-test. The steps are as follows:

  1. Calculate the differences \( d \) between the number of times tempers were lost before and after the course for each participant.
  2. Compute the mean and standard deviation of these differences.
  3. Use the t-distribution to find the test statistic, given the sample size and the mean and standard deviation of the differences.
  4. Compare the test statistic to the critical value from the t-distribution table at the 0.05 significance level to determine if the null hypothesis can be rejected.
Step 1: Calculate Differences

We first calculate the differences \( d \) between the number of times tempers were lost before and after the course for each participant: \[ d = \text{after} - \text{before} = [-1, -1, -2, -4, 1, 1, 0, -2, 1, 1, 0, -1] \]

Step 2: Compute Mean and Standard Deviation of Differences

Next, we compute the mean and standard deviation of these differences: \[ \bar{d} = -0.5833 \quad \text{(mean of differences)} \] \[ s_d = 1.5643 \quad \text{(standard deviation of differences)} \]

Step 3: Calculate the Test Statistic

Using the mean and standard deviation of the differences, we calculate the t-statistic: \[ t = \frac{\bar{d}}{s_d / \sqrt{n}} = \frac{-0.5833}{1.5643 / \sqrt{12}} = -1.2918 \]

Final Answer

\(\boxed{t = -1.292}\)

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