(a)

Mean recovery time for patients with a certain virus is days.

Standard deviation is days.

Sample of  people is selected.

Find the probability of an average recovery time of less than days for a random sample of people.

Here .

Standard deviation of a sample mean is .

Substitute and in the formula.

.

Use the -value formula for a sample mean to find the -value for  . 

-value formula: .

Substitute , and .

.

Using -score table:

The area for is .

Probability of an average recovery time of less than days for a random sample of people is . 

(b)

Sample of people is selected.

Find the probability of an average recovery time is between and days.

Here .

Standard deviation of a sample mean is .

Substitute and in the formula.

Use the -value formula for a sample mean to find the -value for  and

-value formula: .

Substitute , and .

.

Using -score table:

The area for is .

-value for .

Substitute , and .

.

Using -score table:

The area for is .

Probability of an average recovery time is between and days is .

(a) Probability of an average recovery time of less than days for a random sample of people is . 

(b) Probability of an average recovery time is between  and days is .