Question

A study of the total number of minutes spent in the local emergency room revealed that patients spent on average 162.8 minutes in the ER, with a standard deviation of 53.4 minutes. Assume that the times follow a normal distribution. Let X be the processing time of a randomly selected patient.

(a) Describe the probability distribution of X and state its parameters μ and σ.

(b) Find the probability that the processing time of a randomly selected patient is

i. less than 3 hours.

ii. between 140 and 200 minutes.

iii. more than 2 hours.

(c) Find the 80-th percentile for the processing time of a randomly selected patient.

Answer 1

(a) The probability distribution of X is normal distribution with parameter minutes and =53.4 minutes.

(b)

(i) Now ,

; From standard normal probability table

Therefore , probability that the processing time of a randomly selected patient is less than 3 Hours is 0.6255

(ii)

; From standard normal probability table

Therefore , probability that the processing time of a randomly selected patient is between 140 and 200 minutes is 0.4244

(iii) Now ,

; From standard normal probability table

Therefore , probability that the processing time of a randomly selected patient is more than 2 hours is 0.7881

(c)

Now ,

.............(I)

Also , form standard normal probability table ,

..........(II)

From (I) and (II) , we get ,

Therefore , the 80-th percentile for the processing time of a randomly selected patient is 207.6560