A logistics engineer is evaluating the performance of two routing algorithms, Alpha and Beta, by measuring delivery times (in minutes) across 8 test routes. The engineer believes that the mean delivery time for the Beta algorithm is at least 6.8 minutes longer than the mean delivery time for the Alpha algorithm. The recorded times are shown in the table below:
Route12345678Beta52.449.863.551.257.145.960.354.6Alpha45.143.156.444.250.239.053.547.4 \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \text{Route} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \text{Beta} & 52.4 & 49.8 & 63.5 & 51.2 & 57.1 & 45.9 & 60.3 & 54.6 \\ \hline \text{Alpha} & 45.1 & 43.1 & 56.4 & 44.2 & 50.2 & 39.0 & 53.5 & 47.4 \\ \hline \end{array} RouteBetaAlpha152.445.1249.843.1363.556.4451.244.2557.150.2645.939.0760.353.5854.647.4Assuming that the delivery times are normally distributed, test the engineer's belief at the 5% level of significance. State your hypotheses clearly and show all your calculations.