Question

A dishwasher has a mean life of 12 years with an estimated standard deviation of 1.25 years ("Appliance life expectancy," 2013). The life of a dishwasher is normally distributed. Suppose you are a manufacturer and you take a sample of 10 dishwashers that you made.

1. State the random variable.
2. Find the mean of the sample mean.
3. Find the standard deviation of the sample mean.
4. What is the shape of the sampling distribution of the sample mean? Why?
5. Find the probability that the sample mean of the dishwashers is less than 6 years.
6. If you found the sample mean life of the 10 dishwashers to be less than 6 years, would you think that you have a problem with the manufacturing process?   Why or why not?

Answer 1

Answer)

Random variable is life of dishwasher

Mean of the sample mean is same as the population mean = 12

S.d = s.d/√n = 1.25/√10 = 0.39528470752

Shape of the sampling distribution is normal

Because, according to the central limit theorem, sample taken from the normally distributed population is normal

As the data is normally distributed we can use standard normal z table to estimate the answers

Z = (x-mean)/s.d

Z = (6-12)/(1.25/√10) = -15.18

From z table, P(z<-15.18) = 0

Yes if we  found the sample mean life of the 10 dishwashers to be less than 6 years,

There would.be problem

As the probability is extremely small