An investigation of two types of bulldozers showed that 50 failures of one type of bulldozer took on an average 6.8 hours to repair with a standard deviation of 0.85 hours, while 50 failures of the other type of bulldozer took on an average 7.3 hours to repair with a standard deviation of 1.2 hours.

(a) Test the null hypothesis \(\mu_{1}-\mu_{2}=0\) (namely, the hypothesis that on an average, it takes an equal amount of time to repair either kind of bulldozer) against the alternative hypothesis \(\mu_{1}-\mu_{2} eq 0\) at the level of significance, \(\alpha=0.10\).

(b) Using 0.85 and 1.2 as estimates of \(\sigma_{1}\) and \(\sigma_{2}\), find the probability of failing to reject the null hypothesis \(\mu_{1}-\mu_{2}=0\) with the criterion of part

(a) when actually \(\mu_{1}-\mu_{2}=-3\).