Studying the flow of traffic at two busy intersections between 4 P.M. and 6 P.M. (to determine the possible need for turn signals), it was found that on 40 weekdays there were on the average 247.3 cars approaching the first intersection from the south that made left turns while on 30 weekdays there were on the average 254.1 cars approaching the second intersection from the south that made left turns. The corresponding sample standard deviations are \(s_{1}=15.2\) and \(s_{2}=18.7\).

(a) Test the null hypothesis \(\mu_{1}-\mu_{2}=0\) against the alternative hypothesis \(\mu_{1}-\mu_{2} eq 0\) at the level of significance \(\alpha=0.01\).

(b) Using 15.2 and 18.7 as estimates of \(\sigma_{1}\) and \(\sigma_{2}\), find the probability of failing to reject (accepting) the null hypothesis \(\mu_{1}-\mu_{2}=0\) when actually \(\left|\mu_{1}-\mu_{2}\right|=15.6\).