If 12 cows are randomly selected from the very large herd to go on the truck, what is the probability their total weight will be over the maximum allowed 5784?

Question

On the distance planet cowabunga, the weights of cows have a normal distribution with a mean of 462 pounds and a standard deviation of 38 pounds. The cow transport truck holds 12 cows and can hold a maximum weight of 5784. Give answer correct to 4 decimal places.

Answer 1

If the maximum weight is 5784, then the maximum mean weight allowed is 5784/12 or 482. This problem is actually asking what the probability of having a sample of 12 cows whose mean weight is less than or equal o 482.

We calculate the probability by converting to a z-score using the central limit theorem (which is allowed since the cows weights is normally distributed):

z = (x_bar - µ)/(σ/√n) = (482-462)/(38/√12) ≈ 20/10.9697 ≈ 1.82

The probability that the mean is at or below 482 is the same as P(Z ≤ 1.82). Looking this up on a table or using technology you find the probability to be 0.9656.