2025 DHS P2 Q10

In this question you should state the parameters of any distributions that you use.

To regulate the use of its facilities by its paying members, a gym provides a guideline of a maximum usage time of \(90\) minutes per visit per paying member. However, the gym’s records show that the usage time, \(X\) minutes, of its paying members follows the distribution \(\mathrm{N}(95, \sigma^2)\), where \(\sigma\) is the standard deviation. Only \(30\)% of its paying members adhere to the guideline.

1. Show that \(\sigma = 9.53\), correct to \(3\) significant figures.[2]
2. The gym closes at \(11\) pm. John, a paying member, arrives at \(9.15\) pm. Find the probability that he cannot finish his workout.[1]
3. Ten paying members are randomly chosen. Find the probability that the usage time of the ninth paying member is the third and last one that adhere to the guideline.[2]

To grow its membership, the gym allows each paying member to sign up a friend as a trial member. For a limited period, the trial member can use the gym for free. Over time, it is found that the usage time, \(W\) minutes, of its trial members follows the distribution \(\mathrm{N}(85, 80)\).

4. Sketch the distribution of \(X\) and \(W\) on the same diagram.[2]
5. Find the probability that the total usage time spent by two randomly chosen paying members differs from twice the time spent by a randomly chosen trial member by at least \(15\) minutes.[4]

6. State an assumption needed for your calculation in part (e) to be valid.

   [1]