An alphabet of problems:You are hired as consultant for a fleetof small fishing boats which hire out to amateur fishermen. You will consider one day's haul for the entire fleet to be a simple random sample from an appropriate probability distribution. Unless otherwise indicated, let α = 0.05.Suppose the average weight of fishin the area of your fishingoperation is 7.5202 pounds with a standard deviation of 2.5311 pounds. Further, suppose there were 1756 fish in today's haul for the fleet with 547 flounder and with 368 fish weighing more than ten pounds.(a) How likely is it a haul like today's of 1756 fish would weigh more than 13,000 pounds total? [answer: 0.9738](b) Justify the distribution you used in part (a). Why do you not need to assume the weight of fish to be normally distributed?(c) What should today's haul weigh? Justify this precisely in terms of our models. (d) What is the standard deviation for the total weight of a haul of 1756 fish? (e) What is the random variable whose probability distribution is being sampled in part (a)? Verify that it is, indeed, a random vari- able and that a “haul” is a simple random sample from its distribution .(f) Explain how in part (e) “schools of fish” might cause the independence assumption in the definition “simple random sampling from a probability distribution” to fail and why for your operation it's reasonable to think the assumption is approximately valid.(g) The weight of a haul is  where X is the weight of one fish in the haul (sample). You used this in parts (a) and (d). Verify that  is a random variable. What is its distribution? Give formulas for its mean and its variance.(h) A teenage girl goes out into the fleet's fishing waters with her new boat. Her day's haul of four fish weighs a total of 23.2 pounds. The total squared deviations from the mean for her sample is 2.2707 squared pounds. She wonders how much the fish in these waters weigh on average. Based on her catch, what might she conclude?(i) The girl in part (h) returns to her fishing area the next day and decides to stay until she gets one fish. How much will that fish weigh? (j) What two assumptions must the girl in part (h) make? Are they reasonable? How might they fail? (k) What is it that tells us the girl in part (h) seems to be fishing in an area where the fish are smaller than the fish of your fleet? Explain how such a situation might actually arise. (l) What proportion of the fish in the fleet's area of operations weigh more than ten pounds?(m) What proportion of the fish in the fleet's area of operations are flounder?(n) How large a haul would you have to get to estimate the proportion of flounder in the fleet's waters to within 0.14 of a percentage point? Do a worst case analysis because today's haul might be very unrepresentative.(0) The teenage girl in part (h) rented her boat from an old man who said, “Waal, de fish 'round here don' weigh much more'n about five pounds.” Does the girl's catch call the old man's information into question? (p) Each month, you examine a sample of 400 fish from the month's haul for evidence of toxic contamination. How many of the fish in your sample showing evidence of such contamination would cause you to take action to identify and correct the contamination? According to the director of operations, fewer than one contaminated fish in 1000, on average, would be insignificant.(q) What are the chances an amateur fisherman on one of your boats will haul in a fish weighing more than 15 pounds?(r) For your fleet, how much would a haul of 2000 fish weigh on average?(s) What's the probability the typical fish in a haul of 2000 fish would weigh less than seven and a half pounds?(t) For several days, a very large school of fish are in the vicinity of one of your boats. How large a haul would be required to estimate the size of these fish to within one pound? As a rough estimate, assume these fish vary in size pretty much as all the fish in the area of your fleet's operation.(u) In the previous part, why might one think about the z-distribution and why in the last analysis would it not be required? (v) You are watching the weight of the fish in your fleet's waters from season to season to assure satisfaction of the fleet's customers with the quality of fish they catch. If there is ever evidence that the average weight of all fish in the fleet's area of operation has fallen below seven pounds, you will suggest a study to identify any possible cause for a decrease in weight. What should be the criterion for initiating such a study? Assume you monitor the fish each season by weighing a sample of 3500 fish.(w) A year ago there were 601 flounder in a haul of 2144 fish from these waters. Does today's haul support the contention there were fewer flounder in these waters at that time?(x) Each month you examine a sample of 400 fish from the month's haul for evidence of toxic contamination. How much contamination in your sample would cause you to take action to identify and correct the contamination? “Toxic contamination” in this case means more than 0.23 g of the contaminant per fish.(y) How many fish would you have to catch before getting one that weighed more than eight pounds?(z) On one day, one of your boats made a haul of 52 fish. Seven of those fish weighed more than eight pounds. Were there a surprising number of large fish in that day's haul?