Two Sample Tests - Agricultural Statistical Data Analysis Using Stata - page 63

4
t wo S amPle t eStS

Simple statistical tests are available to determine if two means are differ-
ent from one another. Such tests assume that the data are from a normal
distribution, which, of course, is the famous bell-shaped curve. Two
statistics can describe all such distributions, the mean and the variance.
One such statistic that can be used to determine if two means are
different is the Z-test. This statistic does have some limitations and, in
this context, it is rarely used. The primary limitation is the assumption
that the population variance is known. In most cases, the entire popula-
tion is not known. Instead, a sample from the population is used. This
test can be used when sample sizes are large enough, which is seldom
the case in planned experiments. Before the widespread use of comput-
ers, it used to be, as a rule of thumb, that sample sizes greater than 30
from a normally distributed population were sufficient to use the Z-test.
Stata does not supply the Z-test, per se, in the program, but it does
calculate several density functions, one of which is the normal distri-
bution of Z. Using the generalized formula below you can calculate a
Z value and then compare it to the normal (Z) to see if it is significant.

X −µ

σ

o

Z =

n

In this formula, X represents the mean of the value of interest.
μ 0 represents the population mean. The σ value is the population
standard deviation and the n is the sample size. For example, using
the Employee Salaries.dta, you can see how this works. This dataset
consists of salaries for employees in the poultry industry. A random
sample was obtained from a normally distributed population that
consisted of salaries from poultry processing plants and feed mills. To
begin with, let's use the tabstat command to display the means and
standard deviations for this data. Enter

63