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

Answer:

$n=(\frac{1.64(0.85)}{0.25})^2 =31.09$

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

Step-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".

$\bar X$ 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:

$\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$  

The margin of error is given by this formula:

$ME=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$    (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:

$n=(\frac{z_{\alpha/2} s}{ME})^2$   (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:

$n=(\frac{1.64(0.85)}{0.25})^2 =31.09$

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