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,what is the probability that the average weight of the boxes will be less than 84 lbs.?
A)16.87%
B)93.32%
C)43.32%
D)6.68%
E)84.13%

Question

Quizplus · Accepted Answer

The population mean and standard deviation are known, and the sample size is large (n=36), so we can use the Central Limit Theorem to approximate the sampling distribution of the sample mean as normal with a mean of 90 and a standard deviation of 24/sqrt(36) = 4.

To find the probability that the average weight of the boxes will be less than 84 lbs, we can standardize this value using the formula:

z = (x - mu) / (sigma / sqrt(n))

where x is the sample mean we are interested in (84 lbs), mu is the population mean (90 lbs), sigma is the population standard deviation (24 lbs), and n is the sample size (36).

Plugging in the values, we get:

z = (84 - 90) / (24 / sqrt(36)) = -1.5

Using a standard normal table or calculator, we can find that the probability of getting a z-score less than -1.5 is approximately 0.0668, or 6.68%. Therefore, the probability that the average weight of the boxes will be less than 84 lbs is 6.68%, which corresponds to choice D.

A Manufacturing Company Measures the Weight of Boxes Before Shipping