In a manufacturing process, a random sample of 9 manufactured bolts has a mean length of 3 inches with a variance of .09. What is the 90 percent confidence interval for the true mean length of the manufactured bolt?

So on this case the 90% confidence interval would be given by (2.81;3.19)  

Explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

represent the sample mean

population mean (variable of interest)

represent the sample standard deviation

n=9 represent the sample size  

Solution to the problem

The confidence interval for the mean is given by the following formula:

  (1)

In order to calculate the critical value we need to find first the degrees of freedom, given by:

Since the Confidence is 0.90 or 90%, the value of and , and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.05,8)".And we see that

Now we have everything in order to replace into formula (1):

So on this case the 90% confidence interval would be given by (2.81;3.19)    

answer; /// i believe that the correct answer is ///making it easy to work with the organization;

the investment was $14,174.196

explanation:

a lump sum means that the father of the example made one deposit and leave it be.

we need to know the value of a capital that generated at 4.05% per year during 28 years the sum of $48,613.24

now clearing the formula to solve for capital we got:

and we reach the answer investment = $14,174.20