A systems analyst tests a new algorithm designed to work faster than the currently$-$used algorithm. Each algorithm is applied to a group of $56$

sample problems. The new algorithm completes the sample problems with a mean time of $10\mathrm{.}32$

hours. The current algorithm completes the sample problems with a mean time of $10\mathrm{.}51$

hours. Assume the population standard deviation for the new algorithm is $5\mathrm{.}268$

hours, while the current algorithm has a population standard deviation of $5\mathrm{.}279$

hours. Conduct a hypothesis test at the $0\mathrm{.}05$

level of significance of the claim that the new algorithm has a lower mean completion time than the current algorithm. Let $\backslash$mu $1$

be the true mean completion time for the new algorithm and $\backslash$mu $2$

be the true mean completion time for the current algorithm.

Step $1$ of $5$: State the null and alternative hypotheses for the test.