Question

The patient recovery time from a particular surgical procedure is normally distributed with a mean of 4 days and a standard deviation of 1.6 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 5.3 days to recover?

d. What is the probability of spending more than 3.4 days in recovery?

e. What is the probability of spending between 3.8 and 4.3 days in recovery?

f. The 70th percentile for recovery times is days.

Answer 1

Part a)

Part b)

Median = Q2 = 50%

P ( X < ? ) = 50% = 0.5
Looking for the probability 0.5 in standard normal table to calculate critical value Z = 0

0 = ( X - 4 ) / 1.6
X = 4
P ( X < 4 ) = 0.5

Also in Normal distribution Mean = Median = Mode = 4

Part c)

P ( X < 5.3 )
Standardizing the value

Z = ( 5.3 - 4 ) / 1.6
Z = 0.81

Part d)

P ( X > 3.4 ) = 1 - P ( X < 3.4 )
Standardizing the value

Z = ( 3.4 - 4 ) / 1.6
Z = -0.38

P ( Z > -0.38 )
P ( X > 3.4 ) = 1 - P ( Z < -0.38 )
P ( X > 3.4 ) = 1 - 0.352
P ( X > 3.4 ) = 0.648

Part e)

P ( 3.8 < X < 4.3 )
Standardizing the value

Z = ( 3.8 - 4 ) / 1.6
Z = -0.13
Z = ( 4.3 - 4 ) / 1.6
Z = 0.19
P ( -0.13 < Z < 0.19 )
P ( 3.8 < X < 4.3 ) = P ( Z < 0.19 ) - P ( Z < -0.13 )
P ( 3.8 < X < 4.3 ) = 0.5744 - 0.4503
P ( 3.8 < X < 4.3 ) = 0.1241

Part f)

P ( X < ? ) = 70% = 0.7
Looking for the probability 0.7 in standard normal table to calculate critical value Z = 0.52

0.52 = ( X - 4 ) / 1.6
X = 4.832   4.8
P ( X < 4.832 ) = 0.7