Question

question 4 of 6, step 2 of 3
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 0.01 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	7	10	11	8	12	9	8	3	7	11	12
after	5	5	11	8	8	8	6	9	4	6	9	12

step 2 of 3: compute the value of the test statistic. round your answer to three decimal places.
answer

Explanation:

Step1: Calculate the paired - differences

Let \(d_i\) be the difference between the number of times temper was lost after and before the course for each participant.
\[d_1=5 - 6=- 1,d_2=5 - 7=-2,d_3=11 - 10 = 1,d_4=8 - 11=-3,d_5=8 - 8 = 0,d_6=8 - 12=-4,d_7=6 - 9=-3,d_8=9 - 8 = 1,d_9=4 - 3 = 1,d_{10}=6 - 7=-1,d_{11}=9 - 11=-2,d_{12}=12 - 12 = 0\]

Step2: Calculate the mean of the paired - differences \(\bar{d}\)

\(\bar{d}=\frac{\sum_{i = 1}^{n}d_i}{n}\), where \(n = 12\)
\(\sum_{i=1}^{12}d_i=-1-2 + 1-3+0-4-3 + 1+1-1-2+0=-13\)
\(\bar{d}=\frac{-13}{12}\approx - 1.083\)

Step3: Calculate the standard deviation of the paired - differences \(s_d\)

\[s_d=\sqrt{\frac{\sum_{i = 1}^{n}(d_i-\bar{d})^2}{n - 1}}\]
First, calculate \((d_i-\bar{d})^2\) for each \(i\):
For \(d_1=-1\), \((-1+1.083)^2=(0.083)^2 = 0.006889\)
For \(d_2=-2\), \((-2 + 1.083)^2=(-0.917)^2=0.840889\)
...
After calculating \((d_i-\bar{d})^2\) for all \(i\) and summing them up \(\sum_{i = 1}^{12}(d_i-\bar{d})^2\approx22.9167\)
\(s_d=\sqrt{\frac{22.9167}{11}}\approx1.447\)

Step4: Calculate the test - statistic \(t\)

The formula for the paired - t test statistic is \(t=\frac{\bar{d}-\mu_d}{s_d/\sqrt{n}}\), where \(\mu_d = 0\) (under the null hypothesis)
\[t=\frac{-1.083-0}{1.447/\sqrt{12}}\approx - 2.588\]

Answer:

\(-2.588\)