all other conditions being equal explain why the p value obt

all other conditions being equal, explain why the p value obtained from a z test for a population mean will be smaller than from a t test for a population mean (both tests using the same value). Explain why that only with really large sample sizes (greater than a few thousand) will the p value from a z test be approximately the same as the p value from a t test.

Solution

z-test for population mean use population standard deviation and t-test for population mean use sample standard deviation. For small samples, sample standard deviaiton is more variable, that is why t-distribution curve is flatter than z-distribution curve. Since curve of t-distribution is flatter than z-distribution curve for samll samples, so p-value obtained from a z test for a population mean will be smaller than from a t test for a population mean (both tests using the same value).

As the sample sizes increases sample standard deviation becomes closer to population standard deviation. And for large sample sizes sample standard deviation is approximately equal to population standard deviaton. That is for very large sampel size, both t and z distribution has approximately same curve, and so the p value from a z test be approximately the same as the p value from a t test.