The time required for a housekeeping to clean a hotel room for the next customer

Question

The time required for a housekeeping to clean a hotel room for the next customer varies between 20 and 44 minutes and follows the continuous uniform distribution.

a. Calculate the value of f(x).

b. What are the meand and standard deviation for this distribution?

c. What is the probability that the next room will require exactly 27 minutes to clean?

d. What is the probability that the next room will require less than 33 minutes to clean?

e. What is the probability that the next room will require more than 35 minutes to clean?

f. What is the probability that the next room will require between 28 and 41 minutes to clean?

g. What time represents the 70th percentile of this distribution?

Explanation / Answer

a = 20 and b = 44

a)
f(x) = 1/(b-a) = 1/(44 - 20) = 0.0417

b)
mean = (a+b)/2 = (20 + 44)/2 = 32

variance = 1/12*(b-a)^2 = (44 - 20)^2/12 = 48
std. dev. = sqrt(48) = 6.9282

c)
P(X = 27) = f(27) = 0.0417

d)
P(X < 33)
= (33 - 20)/(44 - 20)
= 0.5417

e)
P(X > 35)
= (44 - 35)/(44 - 20)
= 0.375

f)
P(28 <= X <= 41)
= (41 - 28)/(44 - 20)
= 0.5417

g)
(x-a)/(44 - 20) = 0.7
(x - 20)/24 = 0.7
x = 0.7*24 + 20 = 36.8

36.8 i.e. 37 represents the 70th percentile

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