The patient recovery time from a particular surgical procedure is normally distributed with a mean of 3 days and a standard deviation of 1.6 days.

(a) We are informed that a certain surgical procedure's recuperation time for patients is typically distributed. As a result, X is distributed normally.

(b) The median will be equal to the mean because the patient recovery time has a normally distributed distribution. The median is therefore 3 days.

(c) The formula for z score is given as:

z = X - μ / σ

Now, we have the mean μ = 3 days.

Standard deviation, σ = 1.6 days

The observed value, X = 4.2 days

Therefore,

z = ( 4.2 - 3) / 1.6

z = 1.2 / 1.6

z = 0.75

(d) When X = 2.3 days

Then the z score will be:

z = X - μ / σ

z = 2.3 - 3 / 1.6

z = - 0.7 / 1.6

z = - 0.4375

The corresponding area for z = 0.4375 is 0.3336.

In order to determine the likelihood for a period longer than 2.3 days, we must look at the region to the right, which we can locate by deducting 0.3336 from 1.

Therefore, the probability will be:

p = 1 - 0.3336

p = 0.6664

p = 66.64 %

(e) Now, for the probability between X = 3.7 days and 4.2 days.

z₁ = (3.7 - 3) / 1.6 = 0.7/1.6 = 0.4375

z₂ = (4.2 - 3) / 1.6 = 1.2 / 1.6 = 0.75

The area for a z score for 0.4375 is 0.1664 and the area to left for a z score 0.75 is 0.2734.

Then the probability between 3.7 days and 4.2 days will be:

p = 0.2734 - 0.1664

p = 0.107

p = 10.7 %

(f) For the 85th percentile:

For 0.85 the z score from the table is 1.4.

So, z = 1.4

z = X - μ / σ

1.4 = ( X - 3) / 1.6

(1.4)(1.6) = X - 3

2.24 = X -3

X = 2.24 + 3

X = 5.24 days

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The complete question is mentioned below:

The patient recovery time from a particular surgical procedure is normally distributed with a mean of 3 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?

b. What is the median recovery time? __________days

c. What is the Z-score for a patient that took 4.2 days to recover?

d. What is the probability of spending more than 2.3 days in recovery?

e. What is the probability of spending between 3.7 and 4.2 days in recovery?

f. The 85th percentile for recovery times is _________days.