Math Problem Statement
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 participants before completing the anger-management course be Population 1 and let participants after completing the anger-management course be Population 2.
Number of Times Temper Was Lost during a Two-Week Period Before 8 9 7 3 11 5 10 9 10 3 4 5 After 6 5 5 3 8 6 9 10 6 4 5 4
Step 2 of 3 : Compute the value of the test statistic. Round your answer to three decimal places.
Solution
To compute the value of the test statistic for the paired -test, follow these steps:
Step 1: Compute the differences
For each participant, calculate the difference between the "Before" and "After" counts:
Participant | Before | After | Difference () |
---|---|---|---|
1 | 8 | 6 | 2 |
2 | 9 | 5 | 4 |
3 | 7 | 5 | 2 |
4 | 3 | 3 | 0 |
5 | 11 | 8 | 3 |
6 | 5 | 6 | -1 |
7 | 10 | 9 | 1 |
8 | 9 | 10 | -1 |
9 | 10 | 6 | 4 |
10 | 3 | 4 | -1 |
11 | 4 | 5 | -1 |
12 | 5 | 4 | 1 |
Step 2: Calculate the mean () and standard deviation () of the differences
-
Compute :
-
Compute :
-
Compute for each participant, and then sum:
Participant | ||
---|---|---|
1 | 2 | |
2 | 4 | |
3 | 2 | |
4 | 0 | |
5 | 3 | |
6 | -1 | |
7 | 1 | |
8 | -1 | |
9 | 4 | |
10 | -1 | |
11 | -1 | |
12 | 1 |
- Compute :
Step 3: Compute the test statistic
The test statistic for the paired -test is:
Substitute the values:
Final Answer:
The value of the test statistic is:
Would you like me to continue with further steps or interpret the result?
Related Questions:
- How do you determine the critical value for this test?
- What does the -value indicate in the context of this problem?
- How does the level of significance affect the decision to reject the null hypothesis?
- What are the assumptions of a paired -test?
- How could this test be applied to other areas of study?
Tip:
Always check that the differences are approximately normally distributed before performing a paired -test.
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Math Problem Analysis
Mathematical Concepts
Statistics
Hypothesis Testing
Paired t-Test
Formulas
Mean of differences: \( \bar{d} = \frac{\sum d_i}{n} \)
Standard deviation of differences: \( s_d = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n-1}} \)
Test statistic for paired t-test: \( t = \frac{\bar{d}}{s_d / \sqrt{n}} \)
Theorems
Central Limit Theorem (for assuming approximate normality)
t-Distribution for small sample sizes
Suitable Grade Level
College-level Statistics