Test Statistics vs. Z Score

[keep_running]

When do we know to use standard deviation vs. standard error in the denominator for a test stat? I am asking this because the z score allows us to use the standard deviation in the bottom, while other test statistics want us to use the standard error in the denominator..

 

[S2000magician]

If you’re doing an hypothesis test for a single observation (e.g., an hypothesis test for next month’s return on a bond portfolio), then you use the standard deviation.
If you’re doing an hypothesis test for the mean of a set of observations (e.g., an hypothesis test for the average return on an equity portfolio), then you use the standard error.

 

[busprof]

S2000 is correct, but in the case of test of a single observation, the standard deviation and standard error are the same - the standard error is sigma/root, so for a sample of 1, standard error is sigma/1 = sigma.

 

[S2000magician]

busprof wrote:S2000 is correct, but in the case of test of a single observation, the standard deviation and standard error are the same - the standard error is sigma/root, so for a sample of 1, standard error is sigma/1 = sigma.

Perhaps I wasn’t clear enough in my answer.
Suppose that you have a sample of 5 years of monthly returns on a bond fund: 60 data points.
If you’re asked to test the hypothesis that next month’s return on the bond fund is a particular value, you will use the standard deviation of the (60) monthly returns in computing the t-statistic.
If you’re asked to test the hypothesis that the average monthly return on the bond fund is a particular value, you will use the standard error of the (60) monthly returns in computing the t-statistic: the standard deviation of the 60-month sample divided by √60.