Question

A machine used to fill gallon-sized paint cans is regulated so that the amount of paint dispensed has a mean of 120 ounces and a standard deviation of 0.30 ounce You randomly select 40 cans and carefully measure the contents. The sample mean of the cans is 119.9 ounces. Does the machine need to be reset? Explain your reasoning it is that you would have randomly sampled 40 cans with a mean equal to 119.9 ounces, because it within the range of a usual event, namely within of the mean of the sample means

Answer 1

Given that

$\mu=120,\sigma=0.30$

The standard error of mean is

$se=\frac{\sigma}{\sqrt{n}}=\frac{0.30}{\sqrt{40}}=0.0474$

The usual range of values is within 2 standard deviations of mean. Here usual range of values will be

$\mu\pm 2se=120\pm 2\cdot 0.0474=(119.9052,120.0948)=(119.91,120.09)$

Yes.It is unusual that you would have randomly sampled 40 cans with a mean equal to 119.9 ounces because it is not within range of a usual event namely with 2 standard deviations of the mean of the sample means.