Under the mentioned assumptions, the pooled \(t\) test for testing \({H_0}:{\mu _1} - {\mu _2} = {\Delta _0}\) uses the following \(t\) value
\(t = \frac{{\bar x - \bar y - {\Delta _0}}}{{{s_p} \cdot \sqrt {\frac{1}{m} + \frac{1}{n}} }}\)
where the corresponding \(T\) statistic has students distribution with\(m + n - 2\) degrees of freedom, and \({s_p}\)is
\({s_p} = \sqrt {\frac{{(m - 1)s_1^2 + (n - 1)s_2^2}}{{m + n - 2}}} \)
Now use this procedure on the data from exercise \(33\) . The test statistic value is
\(\begin{array}{l}t = \frac{{5.8 - 3.8 - 1}}{{\sqrt {\frac{{(32 - 1) \cdot {{3.2}^2} + (32 - 1) \cdot {{2.8}^2}}}{{32 + 32 - 2}}} \cdot \sqrt {\frac{1}{{32}} + \frac{1}{{32}}} }}\\ = \frac{1}{{51.34 \cdot 0.25}} = 0.078.\end{array}\)
The corresponding \(T\) statistic has \(32 + 32 - 2 = 62\) degrees of freedom and the \(P\) value for testing \({H_0}:{\mu _1} - {\mu _2} = 1\) versus \({H_1}:{\mu _1} - {\mu _2} > 1\) is
\(P = P(T > 0.078) = 0.53\)
which was computed using software. NOTE: the writer of the book probably had some other exercise in mind. The\(P\)value is very high, therefore it strongly suggest to
not reject null hypothesis
