Pete's Powerful Pills (PPP) has branched out and is now promoting a vaccine, which is supposed to prevent statisticitis, a nasty disease afflicting students at the local Evermore University. They are trying to convince the university's health center that the vaccine is worth stocking. The campus health center has agreed to stock and administer the vaccine if it sells well enough, at least = 50 vaccines per week. PPP arranges for the health center to conduct a 36-week trial with the goal of convincing the center to stock the vaccine. Over that period, they found that the center has administered an average, of 55 vaccines per week with a standard deviation s = 6.99.

1. Compute a 95% confidence interval for , the mean number of vaccines the health center will administer per week in the long run. Does it look like the health center will be willing to stock the vaccine? Explain.
2. Perform the hypothesis test using test statistics that PPP will conduct, to convince the health centre that the vaccine is worth stocking. In other words, give the null and alternative hypotheses, both mathematically and in words, and explain your reasoning.
3. Conduct the hypothesis test using p-value and state your real-world conclusions assuming you need to be 95% sure the vaccine is worthwhile before you stock it. Is this consistent with your result from part (ii)?
4. Suppose that PPP makes a $5 profit on every vaccine administered by the health centre. Furthermore, suppose they need to average profits of at least $260 per week for it to be worthwhile to market the vaccine at the school. Can they be 95% (really 97.5%) sure of meeting their goal? Explain. (Hint: Use the CI computed in part (i)).