Attendance at a health club is normally distributed. The average number of peopl

Question

Attendance at a health club is normally distributed. The average number of people visiting the health club at any given time is 100, with a standard deviation of 48.5. The club has a capacity of 200 people. 4. What is the probability the club exceeds capacity? a. b. In the past, club workers have wanted to close early on evenings before holidays when attendance was very low. The current club contract with members states that if attendance is below 5 people, the club has the option of closing early. What is the probability the club will close early?

Explanation / Answer

Given ,’Average number of visitors = m = 100

              Standard deviation of number of visitors = S = 48.5

Let z value corresponding to the probability that number of visitors not to exceed 200 = Z1

Hence,

M + Z1.S = 100

Or, 100 + 48.5.Z1 = 200

Or . 48.5Z1 = 100

Or, Z1 = 2.06

Corresponding probability for Z1 = 0.98030

Hence, probability that the club exceeds capacity i.e. number of visitors exceed 200 = 1 – 0.98030 = 0.0197

Similarly, let the Z value corresponding to probability that number of visitors not to exceed 5 = Z2

Hence,

M + Z2.S = 5

Or, 100 + .48.5.Z2 = 5

Or, 48.5,Z2 = 5 – 100

Or, 48.5.Z2 = - 95

Or, Z2 = - 95/48.5 = - 1.96

Corresponding probability for Z = - 1.96 from standard normal distribution table is 0.025

Hence probability that the club will close early 0.025

PROBABILITY THAT THE CLUB EXCEEDS CAPACITY = 0.0197

PROBABILITY THAT THE CLUB WILL CLOSE EARLY = 0.025

PROBABILITY THAT THE CLUB EXCEEDS CAPACITY = 0.0197

PROBABILITY THAT THE CLUB WILL CLOSE EARLY = 0.025

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