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.

State the random variable.

Find the mean of the sample mean.

Find the standard deviation of the sample mean.

What is the shape of the sampling distribution of the sample mean? Why?

Find the probability that the sample mean of the dishwashers is less than 6 years.

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

A dishwasher has a mean life of 12 years with an estimated standard deviation of 1.25 years .

It is given that the life of a dishwasher is normally distributed.

n = sample size.

State the random variable.

Let X be a random variable which takes the value of life of a dishwasher.

Find the mean of the sample mean.

The mean of sample mean is same as the mean of individual x.

Therefore mean of sample mean = = 12 years

Find the standard deviation of the sample mean.

The formula of standard deviation of a sample mean is as follows:

Where years.

n = sample size = 10

Plug these value in the above formula , we get.

What is the shape of the sampling distribution of the sample mean? Why?

The shape of the sampling distribution of sample mean is symmetric because the sample is comes from the normal distribution. So the distribution of any linear combination of x is also normally distributed.

Find the probability that the sample mean of the dishwashers is less than 6 years.

That we need to find P( < 6)

Let's use excel:

P( < 6) =NORMDIST(6,16,0.395285,1)" = 0

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?

Yes, we think that you have a problem with the manufacturing process. Because the probability of the sample mean of 10 dishwashers to be less than 6 years is approximately equal to 0.