Ultrasound is one of many experimental treatments used for soft tissue injuries. In an experiment to gauge the efficacy of this treatment for knee injuries, seven subjects with recent knee injuries were enrolled in a trial. The range of motion (in degrees) of the joint was measured first. After a prescribed ultrasound treatment, the range of motion was measured again. Table 1 below contains the pre and post treatment responses from each of these subjects.

Table 1: Knee extensions (degrees) before and after ultrasound

Pre 57.5 28.0 29.7 65.1 44.0 51.7 39.4

Post 37.6 52.6 54.0 39.1 57.9 45.7 58.6

Diff A) In the table above, enter the differences (computed Pre - Post) in the empty cells.

B) Based on the sample, the average difference in the pre and post treatment responses is . (2 decimal places)

C) The margin of error for a 99% confidence interval for the true average difference in knee extension before versus after ultrasound treatment is 29.91. Using this margin of error , determine the lower and upper limits for the confidence interval.

Lower Limit: (2 decimal places)

Upper Limit: (2 decimal places)

D) Based on the results of this study, at = 0.01 we can conclude that :

A) The differences (computed Pre - Post) in the table are as follows:

Pre | Post | Diff
---------------------
57.5 | 37.6 | 19.9
28.0 | 52.6 | -24.6
29.7 | 54.0 | -24.3
65.1 | 39.1 | 26.0
44.0 | 57.9 | -13.9
51.7 | 45.7 | 6.0
39.4 | 58.6 | -19.2

B) The average difference in the pre and post treatment responses is calculated as the mean of the differences:

Average difference = (19.9 - 24.6 - 24.3 + 26.0 - 13.9 + 6.0 - 19.2) / 7 = -6.00 (rounded to 2 decimal places)

C) The margin of error for a 99% confidence interval is given as 29.91. Using this margin of error, the lower and upper limits for the confidence interval can be determined as:

Lower Limit = Average difference - Margin of error = -6.00 - 29.91 = -35.91 (rounded to 2 decimal places)
Upper Limit = Average difference + Margin of error = -6.00 + 29.91 = 23.91 (rounded to 2 decimal places)

D) Based on the results of this study, at α = 0.01 (0.01 significance level), we cannot make a conclusion about the true average difference in knee extension before versus after ultrasound treatment since the confidence interval includes zero.

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