A study of the amount of time it takes a mechanic to rebuild the transmission for a 2005 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics are randomly selected, find the probability that their mean rebuild time is less than 8.9 hours.
A) 0.4276
B) 0.9589
C) 0.9608
D) 0.9756

Question

Quizplus · Accepted Answer

The central limit theorem states that the distribution of sample means will be approximately normal, regardless of the shape of the population distribution, if the sample size is large enough. In this case, the sample size is 40, which is large enough to apply the central limit theorem.The mean of the sample means will be equal to the mean of the population, which is 8.4 hours. The standard deviation of the sample means will be equal to the standard deviation of the population divided by the square root of the sample size, which is 1.8 hours / sqrt(40) = 0.28 hours.Therefore, the probability that the mean rebuild time of the 40 mechanics is less than 8.9 hours is equal to the probability that a normally distributed random variable with a mean of 8.4 hours and a standard deviation of 0.28 hours is less than 8.9 hours. This probability can be calculated using a standard normal distribution table or a calculator, and it is equal to 0.9608.

A Study of the Amount of Time It Takes a Mechanic