100 students are waiting for service at the Office of Enrollment. The waiting ti
ID: 3218084 • Letter: 1
Question
100 students are waiting for service at the Office of Enrollment. The waiting time T, in minutes, of an individual student has an exponential distribution with parameter =3 and the waiting times for different students are independent. It can be shown that the total waiting time, X, of the 100 students (in minutes) has a gamma distribution with shape parameter 100 and scale parameter 1/3. (In R this is shape=100, scale = 1/3. In the text this corresponds to = 100, =1/3)
a) What is the expected value of X in minutes? 1
b) What is the variance of X? 2
c) What is the 80th percentile of X? 3
d) What is the probability that 33 X 38? 4
e) What is the probability that the total waiting time is more than 40 minutes? 5
f) What is the probability that the total waiting time is at most 30 minutes? 6
g) What is the probability that X is within 1 standard deviation of its expected value? 7
h) Copy your R script for the above into the text box here.
8
PLEASE INCLUDE R-SCRIPT ABOVE
Explanation / Answer
Given = 100,
=1/3
a) What is the expected value of X in minutes?
Mean X =
=100/3
=33.33 min
b) What is the variance of X?
Variance =2
=100/9
=11.11
c)What is the 80th percentile of X?
80th percentile of X =36.1015
d) What is the probability that 33 X 38
P(33<X<38) =0.9150-0.4733
=0.4417
e) What is the probability that the total waiting time is more than 40 minutes?
P(X>40) =1-P(X<40)
=1-0.9721
=0.0279
f) What is the probability that the total waiting time is at most 30 minutes?
P(X<30)=0.1582
g) What is the probability that X is within 1 standard deviation of its expected value?
P(100/3-10/3 <X<100/3+10/3) =P(30<X<36.6667)
=0.8417-0.1582
=0.6835
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