Determine the Null and Alternative Hypotheses for the Proposed Hypothesis $\mu _ { 1 }$

Determine the null and alternative hypotheses for the proposed hypothesis test.
-A company which designs sports shoes has made an improvement to their popular running shoe. They hope that athletes wearing the new running shoe will be able to run faster over short Distances. They will use a paired sample to determine whether the mean time to run 100 meters for Sprinters wearing the new running shoe is less than the mean time to run the 100 meters for
Sprinters wearing the old running shoe.

A) Let $\mu _ { 1 }$ denote the mean time to run 100 meters for sprinters wearing the new running shoe and let $\mu _ { 2 }$ denote the mean time to run 100 meters for sprinters wearing the old running shoe. The null and alternative hypotheses are $\mathrm { H } _ { 0 } : \mu _ { 1 } > \mu _ { 2 }$ and $\mathrm { H } _ { \mathrm { a } } : \mu _ { 1 } < \mu _ { 2 }$ .
B) Let $\mu _ { 1 }$ denote the mean time to run 100 meters for sprinters wearing the new running shoe and let $\mu _ { 2 }$ denote the mean time to run 100 meters for sprinters wearing the old running shoe. The null and alternative hypotheses are $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ and $\mathrm { H } _ { \mathrm { a } } : \mu _ { 1 } < \mu _ { 2 }$ .
C) Let $\mu _ { 1 }$ denote the mean time to run 100 meters for sprinters wearing the new running shoe and let $\mu _ { 2 }$ denote the mean time to run 100 meters for sprinters wearing the old running shoe. The null and alternative hypotheses are $\mathrm { H } _ { 0 } : \mu _ { 1 } < \mu _ { 2 }$ and $\mathrm { H } _ { \mathrm { a } } : \mu _ { 1 } > \mu _ { 2 }$ .
D) Let $\mu _ { 1 }$ denote the mean time to run 100 meters for sprinters wearing the new running shoe and let $\mu _ { 2 }$ denote the mean time to run 100 meters for sprinters wearing the old running shoe. The null and alternative hypotheses are $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ and $\mathrm { H } _ { \mathrm { a } } : \mu _ { 1 } \leq \mu _ { 2 }$ .

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