Question: A manufacturing firm recruits 45 new sales staff, and sends each on one of two training courses: 25 attend course A, and 20 attend course B. Their sales figures are then collected for the following six months, and the following summary statistics are produced from profits X (in $1, 000s) generated from these sales:

Training course A n₁ = 25 \({{\bar{X}}_{1}}\) =14.9 s₁ = 2.5

Training course B n₂ =20 \({{\bar{X}}_{2}}\) =13.2 s₂ = 3.5

The firm’s manager decides to analyse the data using a two sample t - test. What assumptions is she making? Write down the distributions of \({{\bar{X}}_{1}}\), \({{\bar{X}}_{2}}\) which follow from these assumptions, and, by combining these results, obtain a test statistic for testing the hypothesis that the two training courses generate equal mean profits.