A scientists conducted a study to compare the efficiency of two computational

algorithms that are both developed to solve a complicated optimization procedure. Using

the same initial and stopping setup, these two algorithms were implemented as two different

scripts. To yield fair comparison, both scripts are run on each of the six machines and a

system reboot between the submissions of the two scripts was enforced. The runtimes $($in

seconds$)$ are recorded as follows:

Machine A B

#$16056$

#$25351$

#$34239$

#$45859$

#$56261$

#$64745$

It is safe to assume that the differences in runtime between the two algorithms is approximately normal distributed. The scientist wishes to use this study to test whether the

algorithm B is in fact faster $($than algorithm A$).$

$($a$)(5$ points$)$ State the appropriate setup for the null and alternative hypothesis $($in terms

of $$a and $$b$,$ the mean runtime for algorithm A and B respectively.$)$

$($b$)(15$ points$)$ Find the value of test statistics. $($all the details including intermediate steps

are expected$).$

$($c$)(5$ points$)$ Find the p$-$value for the test $($using the chart to find the range of p$-$value$).$

$($d$)(5$ points$)$ At the $1\%$ level of significance, can you conclude that algorithm B is faster

than algorithm A