#1) A company owning a national chain of health clubs is starting an ad campaign to encourage members to hire one of the clubs’ personal trainers. The ads contend that not only does using a personal trainer lead to more effective use of the clubs’ equipment, but also “increases the member’s motivation to spend more time using the clubs”.

As evidence to support their claim, the company selected a simple random sample of 100 members who had contracted to use one of the clubs personal trainers, and recorded the total number of hours each such member has spent in the club since Sept. 1 of 2019. Similarly the company selected a simple random sample of 100 members who had not used any personal trainer, and recorded the total number of hours each of those members spent in the club since last Sept. 1 .

Some illustrative data values and some summarized results are shown below: 

a) Explain clearly why we need to treat these as “independent samples”, rather than “paired”.

b) Construct a 95% confidence interval for the true difference in population means of “number of hours spent in the club since Sept. 1” for those “YES” and those “NO”. Be sure to summarize/interpret what your interval says verbally (including whether we are/aren’t able to be as much as 95% confident that the population mean for “YES” is greater than the population mean for “NO”).

c) To what extent does this study and its results “allow you to argue” AND/OR “not allow you to argue” that using a personal trainer is the reason for the “YES” group having a higher population mean of “hours spent in the club since Sept. 1” than the “NO” group ? Explain your reasoning.

d) Suggest a variable that you feel would be a good basis for creating “matched pairs” for this study (i.e. where within each pair, 1 person did have a personal trainer and 1 person did not). Explain the rationale for your choice.

Suppose you followed through on your suggestion, formed 100 pairs of members , analyzed the data and constructed a 95% confidence interval for the “Difference in population means” based on those 100 pairs. What would you be able to look at, to see whether your idea (above) for the basis for creating the pairs was a good one (i.e. where could you see “how much benefit” you got from doing the extra effort to create the 100 paired samples) ?