Shower temperature at the Oxnard Health Club showers is regulated automatically.

Question

Shower temperature at the Oxnard Health Club showers is regulated automatically. The heater kicks in when the temperature falls to 97 degree F and shuts off when the temperature reaches 106 degree F Water temperature then falls slowly until the heater kicks in again. At a given moment, the water temperature is a uniformly distributed random variable U(97, 106). (a) Find the mean temperature. (Round your answer to 1 decimal place.) (b) Find the standard deviation of the temperature. (Round your answer to 4 decimal places.) (c) Find the 75th percentile for water temperature. (Round your answer to 2 decimal places.)

Explanation / Answer

This is a uniform distribution problem with

a = 97 and

b = 106

(a)

mean = 1/2(a+b) = 1/2 * (97+106) = 101.5

(b)

variance = 1/12 * (b-a)^2 = 6.75

std. dev. = sqrt(variance) = 2.5981

(c)

P(X) = d – c / b – a.

Here P(X) = 0.75 and c=a, find d

0.75 = (d - 97)/(106-97)

0.75*9 = d - 97

d = 103.75

hence 103.75 degrees represent the 75th percentile

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