Question

Suppose that a random sample of size 64 is to be selected from a population with mean 40 and standard deviation 5.

a) What is the approximate probability that x will differ from μ by more than 0.8? (Round your answer to four decimal places.)

Answer 1

Given mean = 40

Standard deviation of the sampling distribution = / n

= 5/ 64

= 5/8

we need to calculate the z-values for x= 40+0,8 and 40-0.8

Z-value = (x - ) / / n

Z-value at (x = 40.8) =  (40.8 - 40) / (5/8)

= 0.8 / 0.625

= 1.28

Z-value at (x = 39.2) =  (39.2 - 40) / (5/8)

= -0.8 / 0.625

= -1.28

To get the probabilities we need to get the area to left of these z-values from the respective z-tables attached below

the area to the left of z-score 1.28 can be found from the positive z-score table attached below. The value is 0.89973

the area to the left of z-score -1.28 can be found from the negative z-score table attached below. The value is 0.10027

To get the probability between +1.28 and -1.28 we should subtract 0.10027 from 0.89973 since both are respresenting the area to the left

So approximate probability that x will differ from by more than 0.8 = 0.89973 - 0.10027

= 0.79946

= 0.7995 (rounded to 4 decimals)