You work for a popular pizzeria in suburban Rochester that has recently considered providing a guarantee on delivery times (e.g.,"Guaranteed delivery with

in 50 minutes,or your pizza is free!"). However, you are quite concerned about just how many pizzasyou will be giving away for free under such a policy. You shift managerclaimsthat delivery times are uniformly distributed between 30 minutes and an hour, but you are not so sure about this. Luckily, you have been recording the actual delivery times for the past several years and have assembled a dataset with 1400 observed actual delivery times (DeliveryTimes.xlsx).

• sample average 45.1
• sample standard deviation 2.47
• sample variance of delivery time 6.11

1. Suppose you assume instead that delivery times are normally distributed (rather than uniform) with the mean and standard deviation you found in part a). Using the estimates from part a), construct an interval (cut-off times) into which you expect 95% of delivery times to fall. Repeat this exercise, but replace 95% with 80%. How do these intervals compare with what you would have concluded had you justassumedthat delivery times were distributed uniformly between 30 and 60 minutes? A formal comparison is required here (i.e., construct intervals under the uniform distribution assumption and discuss).
2. Note: Q1(c) is not asking you to construct aconfidence intervalfor thepopulation mean. Q1(c) is asking you to construct an interval inthe normal (and uniform) distribution(recall Q1(e) in Assignment 1?).
3. Continuing to assume a normal distribution, what is the probability that a given pizza is delivered in 45 minutes or less? How about 40 minutes or less? 50 minutes or more? Compare these answers to what you would conclude under the uniform assumption (i.e., repeat this exercise with a uniform distribution).
4. If you were to implement a policy of only charging for pizzas that are delivered in under 50 minutes, would it matter which distribution was the correct one? Why

or why not? (Note: you do not need to compute anything here, just answer the question from an intuitive standpoint).

• Again using the information from part a), construct a 95%confidence intervalfor the population mean. How does this compare to the 95% interval you constructed in part c (for the Normal case)? If it is different, what is the reason for this?

1. Would theconfidence intervalyou constructed in part f) change if you assumed the population distribution was uniform instead of Normal? Why or why not?
2. Using the information from part a), test the null hypothesis that the population average delivery time is 50 minutes against the two-sided alternative hypothesis that it is not 50 minutes at both the 1% and 5% levels. What do you conclude?