The diameter of ball bearings are known to be normally distributed with unknown mean and variance. A random sample of size 25 gave a mean of 2.5 cm. The 95% confidence interval had length 4 cm. Then

(a) The sample variance is 4.86.
(b) The sample variance is 26.03.
(c) The population variance is 4.84.
(d) The population variance is 23.47. (e) The sample variance is 23.47.

And the margin of error is given by:

And the sample variance is

So then the correct option would be:

(e) The sample variance is 23.47.

Step-by-step 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)

s represent the sample standard deviation

n=25 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 0.95 or 95%, 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(0.025,24)".And we see that

The lenght of the interval is given by:

And the margin of error is given by:

And solving for the deviation we got:

And the sample variance is

So then the correct option would be:

(e) The sample variance is 23.47.

24

step-by-step explanation:

this is false ^.^ lol