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:[tex]n=(\frac{1.64(0.85)}{0.25})^2 =31.09 [/tex]So the answer for this case would be n=32 rounded up to the nearest integerStep-by-step 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".[tex]\bar X[/tex] represent the sample mean for the sample  [tex]\mu[/tex] population mean (variable of interest)[tex]\sigma=0.85[/tex] represent the sample standard deviationn represent the sample size (variable of interest)  The confidence interval for the mean is given by the following formula:[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex]   The margin of error is given by this formula:[tex] ME=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex]    (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:[tex]n=(\frac{z_{\alpha/2} s}{ME})^2[/tex]   (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 [tex]z_{\alpha/2}=1.64[/tex], replacing into formula (2) we got:[tex]n=(\frac{1.64(0.85)}{0.25})^2 =31.09 [/tex]So the answer for this case would be n=32 rounded up to the nearest integer