Statistics Normal Distribution

Question

The patient recovery time from a particular surgical procedure is normally distributed with a mean of 3 days and a standard deviation of 1.5 days. Let X be the recovery time for a randomly selected patient. Round all answers to 4 decimal places where possible.

a. What is the distribution of X? X ~ N(,)

b. What is the median recovery time? _______-days

c. What is the Z-score for a patient that took 4.5 days to recover?

d. What is the probability of spending more than 3.7 days in recovery?

e. What is the probability of spending between 2.1 and 3 days in recovery?

f. The 75th percentile for recovery times is _______days.

Answer 1

a) N(3,1.5)

b) since normal distribution median = mean = 3

c) z = (4.5 - 3)/1.5 = 1

d) x number of days to recover:

P(x > 3.7) = P( z > (3.7 - 3)/1.5) = P(z > 0.47) = P(z < - 0.47) = 0.3192

e) P(2.1 < x < 3) = P( (2.1 - 3)/1.5 < z < (3 - 3)/1.5) = P(-1.35 < z < 0) = P(z < 0) - P(z < -1.35) = 0.5 - 0.0885 = 0.4115

f) z for 75th percentile is 0.675

0.675 = (x - 3)/1.55

x = 0.675 * 1.55 + 3 = 4.0125