Question

A normal population has a mean of 57 and a standard deviation of 14. You select a random sample of 16.

Round to 4 decimal places.

a.	33% of the time, the sample average will be less than what specific value?

Value

b.	33% of the time, the value of a randomly selected observation will be less than h. Find h.

h

c.	The probability that the sample average is more than k is 22%. Find k.

k

Answer 1

Given that,

mean = = 57

standard deviation = = 14

n = 16

= = 57

= / n = 14 / 16 = 3.5

Using standard normal table,

P(Z < z) = 33%

= P(Z < z) = 0.33  

= P(Z < -0.44) = 0.33

z = -0.44

Using z-score formula  

= z * +

= -0.4399 * 3.5 + 57

= 55.4604

b) Using z-score formula,

h = z * +

h = -0.4399 * 14 + 57

h = 50.8414

c) Using standard normal table,

P(Z > z) = 22%

= 1 - P(Z < z) = 0.22  

= P(Z < z ) = 1 - 0.22

= P(Z < z ) = 0.78

= P(Z < 0.7722 ) = 0.78  

z = 0.7722

Using z-score formula  

k = z * +

k = 0.7722 * 3.5 + 57

k = 59.7027