Question

A paint manufacturer uses a machine to fill gallon cans with paint​ (1 galequals128 ​ounces). The manufacturer wants to estimate the mean volume of paint the machine is putting in the cans within 0.5 ounce. Assume the population of volumes is normally distributed.​(a) Determine the minimum sample size required to construct a 90​% confidence interval for the population mean. Assume the population standard deviation is 0.80 ounce.​(b) The sample mean is 127 ounces. With a sample size of 8​, a 90​% level of​ confidence, and a population standard deviation of 0.80 ​ounce, does it seem possible that the population mean could be exactly 128 ​ounces? Explain.

Answer 1

a) To determine the minimum sample size we need to use the formula shown in the picture 1.

E is the margin of error, which is the distance from the limits to the middle (the mean) of the confidence interval. This means that we have to divide the range of the interval by 2 to find this distance.

E = 0.5/2 = 0.25

Now we apply the formula

n = (1.645*0.80/0.25)^2 = 27.7 = 28

The minimum sample size would be 28.

b) To answer the question we are going to make a 90% confidence interval. The formula is:

(μ - E, μ + E)

μ is the mean which is 127. The formula for E is shown in the picture.

E = 0.80*1.645/√8 = 0.47

(126.5, 127.5)

This means that the true mean is going to be contained in this interval 90% of the time. This is why it doesn't seem possible that the population mean is exactly 128.