A study of the amount of time it takes a mechanic to rebuild the transmission for a 1992 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.77 hours. Assume that a random sample of 40 mechanics is selected and the mean rebuild time of the sample is computed. Assuming the mean times are normally distributed, what percentage of sample means are greater than 7.7 hours?

Question

A study of the amount of time it takes a mechanic to rebuild the transmission for a 1992 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.77 hours. Assume that a random sample of 40 mechanics is selected and the mean rebuild time of the sample is computed. Assuming the mean times are normally distributed, what percentage of sample means are greater than 7.7 hours?

A. 0.62%
B. 34.46%
C. 65.54%
D. 99.38%
My answer ic C

Answer 1

To solve this problem, we need to use the concept of the sampling distribution of the sample mean.

The sampling distribution of the sample mean is the distribution that represents the possible values that the sample mean can take for different samples of the same size from a population.

Given that the population mean is 8.4 hours and the population standard deviation is 1.77 hours, we can use these values to calculate the standard deviation of the sampling distribution, also known as the standard error. The standard error (SE) can be calculated using the formula:

SE = population standard deviation / square root of sample size

In this case, the sample size is 40. So, the standard error is:
SE = 1.77 / sqrt(40) = 1.77 / 6.32 = 0.2804

Now, we can use the z-score formula to calculate the z-score for the given value of 7.7 hours:
z = (x - population mean) / SE

Plugging in the values, we get:
z = (7.7 - 8.4) / 0.2804 = -0.714

To find the percentage of sample means greater than 7.7 hours, we need to find the area under the standard normal distribution curve to the right of the z-score -0.714.

Using a standard normal distribution table or a statistical calculator, we find that the percentage of values to the right of -0.714 is approximately 0.4816.

However, we are interested in the percentage of sample means that are greater than 7.7 hours, so we need to subtract this value from 1:
1 - 0.4816 = 0.5184

Therefore, the percentage of sample means greater than 7.7 hours is approximately 51.84%.

The correct answer is not listed. Please check your calculations again.

Answer 2

I got 65.38% using David Lane's website

http://davidmlane.com/hyperstat/z_table.html