# Z-test vs. T-test

The average return on stocks from 1926-1997 was 17.7%; the standard error of the sample was 33.9%. What is the 95% confidence interval for the return on stocks for any given year? The book is saying that I should use the Z-test, but I thought we use the T-test when we have an unknown variance (i.e. the standard deviation of the sample) as opposed to the population variance, where we can use Z. I understand that with a large sample size (in this case, 72, which is much greater than the minimum large sample size of 30) the Z-test is accecptable, but the T-test is preffered. Is there any particular reason I should have used the Z-stat in the case?

How do you infer that the population for which the sample is drawn, has unknown variance? Though it has been given that standard error of the sample ( i.e. the standard deviation of the distribution of sample means) is 33.9%; it is insufficient to conclude that the population variance is unknown. Recall from Schwesers Notes that standard error can be calculated as follow:

1. Standard error, σx̄ = σ/√n if population variance is known
2. Standard error, sx̄ = S/ √n if population variance is unknown

Cheers,

Ernest

It’s simpler (no need to worry about the degrees of freedom), and it’s reasonably accurate. The 95%, 2-tailed critical value for a Z statistic is 1.96. The 95%, 2-tailed critical for a t-statistic with 71 degrees of freedom lies between 1.98 (120 dof) and 2.00 (60 dof); call it 1.99. Close enough to 1.96 to call it even.