Question

The shape of the distribution of the time required to get an oil change at a 20​-minute ​oil-change facility is unknown.​ However, records indicate that the mean time is 21.4 minutes, and the standard deviation is 4.5 minutes. ​(b) What is the probability that a random sample of n =35 oil changes results in a sample mean time less than 20 ​minutes? The probability is approximately _______. ​(Round to four decimal places as​ needed.)

Answer 1

Here n = 35 ( > 30 ), which is sufficient for use of central limit theorem.

By central limit theorem , sample mean follow normal distribution with mean = = 21.4

and standard deviation =

Here we want to find P( < 20)

Z score corresponding to 20 is as follow:

So we want to find P( Z < -1.84)

From z table we get  P( Z < -1.84) = 0.0329

Look the following image.

So final answer is 0.0329