Statistics Question

let X denote the scores due to  brightness discrimination,

& Y denote the scores due to pattern discrimination.

sample size, n = 10

Assuming that the scores are normally distributed. To test whether both the tasks were easier to learn or one easier than the other a paired t-test would be used. The hypothesis of no difference is

    $$\begin{gathered} H_0: {\mu }_x={\mu }_y \\ {H}_1: {\mu }_x\ne {\mu }_y \end{gathered}$$

Under the null hypothesis, the test statistic is

       $t=\frac{\overline{d}}{S/\sqrt{n}}~t_{n-1}$

where $\overline{d}=\frac{1}{n}\sum _i{d}_i ;d_i=x_i-y_i, i = 1, 2, ..., n$

 &     $S=\sqrt{\frac{1}{n-1}\sum _i{(d}^{}}$_i-d¯)2

Calculations for the test statistic

    $\overline{d}=\frac{1}{n}\sum _i{d}_i=\frac{1}{10}\times 74=7.4$

  $S=\sqrt{\frac{1}{n-1}\sum _i{(d}^{}}$_i-d¯)2    = 19×312.4  = 5.89