Is this the formula used to acquire the test statistic for both tests? test statistic = (sample statistic-hypothesized value)/(standard error of the sample statistic)

Pretty much. Here is how I look at it: Z-test is for a known pop’n variance so the test statistic = (sample statistic-hypothesized value)/(pop’n variance / root of n) t-test is for an UNKNOWN pop’n variance so the test statistic = (sample statistic-hypothesized value)/(sample variance / root of n)

for z test, are you assuming you DO know the population variance but DO NOT know the population mean? i assume so, as this is the only thing that makes sense to me if you are using standard “error”