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.
355 exam-style questions on OCR (MEI) A Level Maths 2.5 Statistical Hypothesis Testing, covering 2.5.1 Process and language of hypothesis testing, 2.5.2 When to apply 1-tail and 2-tail tests, 2.5.3 Significance level and incorrect rejection, 2.5.4 Null and alternative hypotheses (binomial), 2.5.5 Conduct a binomial hypothesis test, 2.5.6 Critical and acceptance regions (binomial), 2.5.7 Distribution of the sample mean (A-level only), 2.5.8 Hypothesis test for a single mean (A-level only), 2.5.9 Critical and acceptance regions (mean) (A-level only), 2.5.10 Correlation as closeness to a straight line (A-level only), 2.5.11 Inference using a correlation coefficient (A-level only), and 2.5 Statistical Hypothesis Testing. Each one has a worked solution and a mark scheme showing where the marks go.