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Many STA404 students are having problems in data analysis using a statistical software. A lecturer who teaches statistics for the STA404 course offered a two-day workshop on this topic. The following table gives the test scores of seven STA404 students before and after they attended the workshop: Before 56 69 48 74 65 71 58 After 62 73 44 85 71 70 69 Construct a 95% confidence interval for the difference in means of the scores. Based on the interval, can we conclude that the two-day workshop has improved the students' scores?

Question

Many STA404 students are having problems in data analysis using a statistical software. A lecturer who teaches statistics for the STA404 course offered a two-day workshop on this topic. The following table gives the test scores of seven STA404 students before and after they attended the workshop:

Before 56 69 48 74 65 71 58 After 62 73 44 85 71 70 69

Construct a 95% confidence interval for the difference in means of the scores. Based on the interval, can we conclude that the two-day workshop has improved the students' scores?

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Solution

To construct a 95% confidence interval for the difference in means of the scores, we first need to calculate the mean and standard deviation of the differences between the "before" and "after" scores for each student.

Here are the steps:

  1. Calculate the differences between the "before" and "after" scores for each student:

    62-56 = 6 73-69 = 4 44-48 = -4 85-74 = 11 71-65 = 6 70-71 = -1 69-58 = 11

  2. Calculate the mean of these differences:

    (6+4-4+11+6-1+11)/7 = 33/7 = 4.71 (approx)

  3. Calculate the standard deviation of these differences:

    sqrt[((6-4.71)^2 + (4-4.71)^2 + (-4-4.71)^2 + (11-4.71)^2 + (6-4.71)^2 + (-1-4.71)^2 + (11-4.71)^2)/6] = 5.77 (approx)

  4. Calculate the standard error of the mean:

    5.77/sqrt(7) = 2.18 (approx)

  5. Calculate the 95% confidence interval for the difference in means:

    4.71 ± (2.18*2.447) = 4.71 ± 5.33

So, the 95% confidence interval for the difference in means is (-0.62, 10.04).

Since this interval includes zero, we cannot conclude that the two-day workshop has improved the students' scores at the 95% confidence level.

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