a$)$ A compary has develaped a training program for its entering emplayees because they have become concerned with the results of the sik$-$month employee review. They hape that the training program can result in better sixMmonth reviews. Each trainee constitutes a "pair", the entering score the employee recelved when first entering the firm and the score given at the sixmonth review. The difference in the two scores were calculated for each employee and the means for before and after the training program was calculated. The sample mean before the training program was $20\mathrm{.}4$ and the sample mean after the training program was $23\mathrm{.}9.$ The standard deviation of the differences in the two scores across the $20$ emplopees was $3\mathrm{.}8$ points. Test at the $10\%$ significance level the nul hypothesis that the two population means are equal against the alternative that the training program helps improve the employees' scores.

$(3)$