SECTION $6$: Hypothesis Test for Mean Based on Two Samples $(30$ points total, $6$ points for each part$)$

Imagine you are an engineer at a certain airline, and you want to investigate whether the mean arrival time delay for your airline is better than the mean arrival delay time for a competitor airline. You are interested in evaluating this claim using a hypothesis test method. Assume a significance level of $0\mathrm{.}05.$

You will select two airlines at random for this analysis. Please review the Bureau of Transportation Statistics list of airline codes $($USDOT$).$Links to an external site.$$

Use the first letter of your first name to select the first airline. Select the name of the airline that begins with the letter corresponding to the first letter of your first name. If there is no matching airline name corresponding to the first letter of your first name, then pick the name of the airline that most closely matches. For example, if your first name is James, you can select $$JetBlue Airlines.$$ If your first name is Charles, you can select $$Delta Airlines.$$ Consider this to be the airline where you are employed.$$

Use the first letter of your last name to select the second airline. Select the name of the airline that begins with the letter corresponding to the first letter of your last name. If there is no matching airline name corresponding to the first letter of your last name, then pick the name of the airline that most closely matches. For example, if your last name is Anderson, you can select $$American Airlines.$$ If your last name is Rogers, you can select $$Southwest Airlines.$$ Consider this to be the competitor airline.

a$)$ Set up the null and alternative hypotheses. Be sure to set this up using H$0$ and H$1$ notation and use correct symbols for the population parameter of interest.

b$)$ Calculate the test statistic. $$

c$)$ Calculate the corresponding p$-$value.

d$)$ Clearly state your decision as to whether you $$reject the null hypothesis$$ or $$do not reject the null hypothesis.$$ Explain the rationale for your decision.

e$)$ Compose an email to the CEO of your airline explaining your findings. Describe your test methodology and formulate a concluding statement regarding whether the mean arrival time delay for your airline is better than the mean arrival delay time for a competitor airline. Explain the rationale for your decision in language that the CEO can understand and assume the CEO does not have a statistical background.