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6.5.3 t-Test: Two Samples Unequal Variances
A very similar test, but one that does not explicitly consider the variance of the
population to be known ,isthe t-Test . It is reserved for small samples, less than 30
observations, although larger samples are permissible. The lack of knowledge of
a population variance is a very common situation. Populations are often so large
that it is practically impossible to measure the variance or standard deviation of the
population, not to mention the possible change in the population’s membership. We
will see that the calculation of the t-statistic is very similar to the calculation of the
z-statistic .
6.5.4 Do Texas Prisoners Score Higher Than Texas
Non-Prisoners?
Now, let’s consider a second, but equally important question that EB will want to
answer—Is it possible that women prisoners, regardless of the state afﬁliation, nor-
mally score higher than others in the population, and that training is not the only
factor in their higher scores? If we ignore the possible differences in state (SC or
TX) afﬁliation of the prisoners for now, we can test this question by performing
a test of hypothesis with the Texas data samples and form a general conclusion.
Why might this be an important question? We have already concluded that there
is a difference between the mean score of SC prisoners and that of the SC non-
prisoners. Before we attribute this difference to the special training provided by
EB, let us consider the possibility that the difference may be due to the afﬁli-
ation with the prison group. One can build an argument that women in prison
might be motivated to learn and achieve, especially if they are soon likely to rejoin
the general population. As we noted above, we will not deal with state afﬁliation
at this point, although it is possible that one state may have higher scores than
another.
To answer this question we will use the t-Test: Two Samples Unequal
Variances in the Data Analysis tool of Excel. In Exhibit 6.4 we see the dialog
box associated with the tool. Note that it appears to be quite similar to the z-Test,
except that rather than requesting values for known variances, the t-Test calculates
the sample variances and uses the calculated values in the analysis. The results of
the analysis are shown in Table 6.4, and the t-statistic indicates that we should reject
the null hypothesis that the means are the same for prisoners and non-prisoners.
This is so because the –2.53650023 (cell B9) is less than the negative of the criti-
cal two-tail t-value, –1.994435479 (negative of cell B13). Additionally, we can see
that the p-value 0.013427432 (cell B12) is <
(0.05). We therefore conclude that
alternative hypothesis is true—there is a difference between the mean scores of the
prisoners and non-prisoners. This could be due to many reasons and might require
further investigation.
= α
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