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One of the fastest growing businesses in the USE is restaurant chains with delivery service. An investor specialized in food scenes is looking for the

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One of the fastest growing businesses in the USE is restaurant chains with delivery service. An investor specialized in food scenes is looking for the next big delivery chain. She narrows down her search to two Shawarma shops: one called Shawarma Avenue (A) and another one called Shawarma Boulevard (B). To check which shop is more successful in delivering on time, her team orders 30 Shawarmas from each shop on a Saturday night (60 orders total). Shop A was on time in 85% of their deliveries and shop B was on time in 90% oftheirs. Consider that the true probability of on-time delivery is 1),; = 0.9 and p3 = 0.85. Assume also that n=30 is large enough for sampling distributions to be approximately normal and that delivery times for each order are independent from one another. a) What is the expected proportion of on-time deliveries in samples of 30 orders from A and B? (5 points) b) What is the expected variance of proportions of on-time deliveries in samples of 30 orders from A and B? (5 points) c) What is the probability that 85% of orders or fewer from shop A are delivered on time? (3 points) d) What is the probability that 90% of orders or more from shop B are delivered on time? (3 points) The investor fears that, on this particular Saturday, B's deliveries were more often on time just by chance. Let us construct a variable that measures the proportion of times that B will be on-time more often than A. First, we define 13.4 as the random variable measuring the proportion of times that A's deliveries were on time. Second, we define 133 as the random variable measuring the proportion of times that B's deliveries were on time. Lastly, we define a random variable called X such that X = p3 m. e) What is the expected value of X? (3 points) f) What is the expected variance and expected standard deviation of X? (3 points) g) What is the proportion oftimes in which the shop B will be on time more often than shop A just by chance, leading the investor to the wrong conclusion about the efficiency of these shops? (3 points)

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