Question

Paint manufacturer uses a machine to fill 1- gallon cans with paint- they want to estimate the mean volume of paint the machine is putting in the cans within 0.25 of an ounce. Determine the minimum sample size needed to construct a 90% confidence interval for the population mean. Assume the population standard deviation is 0.85 ounces.

Answer 1

`n=((1.64(0.85))/(0.25))^2 =31.09`

So the answer for this case would be n=32 rounded up to the nearest integer

Explanation:

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".

`\bar X` represent the sample mean for the sample

`\mu` population mean (variable of interest)

`\sigma=0.85` represent the sample standard deviation

n represent the sample size (variable of interest)

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

`\bar X \pm z_(\alpha/2)(\sigma)/(√(n))`

The margin of error is given by this formula:

`ME=z_(\alpha/2)(\sigma)/(√(n))` (1)

And on this case we have that ME =0.25 and we are interested in order to find the value of n, if we solve n from equation (1) we got:

`n=((z_(\alpha/2) s)/(ME))^2` (2)

The critical value for 90% of confidence interval now can be founded using the normal distribution. And in excel we can use this formla to find it:"=-NORM.INV(0.05,0,1)", and we got
`z_(\alpha/2)=1.64`, replacing into formula (2) we got:

`n=((1.64(0.85))/(0.25))^2 =31.09`

So the answer for this case would be n=32 rounded up to the nearest integer