In parts of the eastern United States, whitetail deer are a major nuisance to farmers and homeowners, frequently damaging crops, gardens, and landscaping. A consumer organization arranges a test of two of the leading deer repellents $\mathrm{A}$ and $\mathrm{B}$ on the market. Fifty-six unfenced gardens in areas having high concentrations of deer are used for the test. Twenty-nine gardens are chosen at random to receive repellent $\mathrm{A}$, and the other 27 receive repellent $\mathrm{B} .$ For each of the 56 gardens, the time elapsed between application of the repellent and the appearance in the garden of the first deer is recorded. For repellent $\mathrm{A}$, the mean time is 101 hours. For repellent $\mathrm{B}$, the mean time is 92 hours. Assume that the two populations of elapsed times have normal distributions with population standard deviations of 15 and 10 hours, respectively.
a. Let $\mu_{1}$ and $\mu_{2}$ be the population means of elapsed times for the two repellents, respectively. Find the point estimate of $\mu_{1}-\mu_{2}$.
b. Find a $97 \%$ confidence interval for $\mu_{1}-\mu_{2}$.
c. Test at the $2 \%$ significance level whether the mean elapsed times for repellents $\mathrm{A}$ and $\mathrm{B}$ are different. Use both approaches, the critical-value and $p$ -value, to perform this test.