Can someone explain how this answer was found?

Question

A manufacturing company measures the weight of boxes before shipping them to the customers. If the box weights have a population mean and standard deviation of 90 lbs. and 24 lbs. respectively, then based on a sample size of 36 boxes, the probability that the average weight of the boxes will be less than 84 lbs. is:

A) 16.87%

B) 93.32%

C) 43.32%

*D) 6.68%

E) 84.13%

Thus, D is the correct answer.

Answer 1

You are given a mean of μ = 90.  Compute the standard deviation as

σ = s / √n

σ = 24 / √36

σ = 24 / 6

σ = 4

Remember that the standard deviation for the distribution is going to be the estimated standard deviation of the sample divided by the square root of the sample size.  

Now you can find the z-score for 84 lbs as

z = (x - μ) / σ

z = (84 - 90) / 4

z = -6 / 4

z = -1.5

Use a table to find the area below the normal curve to the left of z = -1.5

P(z < -1.5) = .0668

Multiply this by 100 to get the probability in percent.  

.0668 * 100 = 6.68% which is answer D.

Hope this helps you!