The Berg Balance Scale (BBS), named after its developer Katherine Berg, is a popular clinical test of a person's static and dynamic balance abilities. It is scored on a scale of 0 to 56; the higher the value, the better the score. The cut-off score is 45, which means that a score less than 45 indicates a greater risk of falling. The file BBS_Older contains data for 20 older patients at a hospital:

The researchers then collect a new set of data, this time for 20 younger patients. Their findings are stored in the file BBS_Younger :

a. Assuming that the population variances from both researches are equal, is there evidence of a difference in the mean BBS value between the two age groups? (Use \(\alpha=0.05\).)

b. Determine the \(p\)-value in (a) and interpret its meaning.

c. In addition to equal variances, what other assumption is necessary in (a)?

d. Construct and interpret a 95\% confidence interval estimate of the difference between the population means for the two age groups.

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