Question

“PhysicalFun” is a physical center which opens 24 hours per day. Each member of “PhysicalFun” has an access card. Members would be charged in a weekly basic (Sunday 12:00a.m. to Saturday 11: 59 p.m.) according to the accumulated number of hours of facilities used within a week. The weekly basic charge is $150 with an additional charge of $30 per hour. According to the company record, the number of hours a member spends in “PhysicalFun” in a week is normally distributed with mean 15 hours and standard deviation 2.8 hours.

(a) What are the average and standard deviation of weekly charge of a member?

(b) Suppose the middle 85% of weekly charge of a member is denoted by $(L1, L2). Find the values of L1 and L2.

(c) The senior management suggests fixing the weekly charge per member at $700. Assume the number of hours a member spends in “PhysicalFun” would not be changed due to the change of the weekly charge calculation method. What proportion of members would pay more money than the original system?

(d) How many hours does a member spend in “PhysicalFun” in a week so he / she would be beneficial by the new system?

Answer 1

Let X denote the number of hours a member spend in "Physical fun" in a week

Mean of X, = 15

Standard deviation of X, = 2.8

Weekly charge per member, Y = $150 + $30*X

(a) Average weekly charge of a member, E(Y) = E(150 + 30X)

= 150 + 30*15 = $600

Standard deviation of weekly charge of a member = 30*2.8 = $84

(b) Corresponding to middle 85% charges, the z value range are -1.44 to 1.44

Thus, L1 = 600 - 1.44*84 = $479.04

L2 = 600 + 1.44*84 = $720.96

(c) For weekly charge, i.e Y to be $700, X would be 18.33

Thus, proportion of members who would pay more money than the original system = P(X > 18.33)

= P{Z > (18.33 - 15)/2.8}

= P(Z > 1.19) = 0.1170 = 11.7%

(d) The member should spend atleast 18.34 hours so he/she would be beneficial by the new system