BATES ECO-RESORT (48 pts) Norman Bates has just inherited $40,000 from an acquaintance who recently passed away, and he is considering using the money to renovate his namesake motel outside Phoenix, Arizona. At a recent hoteliers' convention, he attended several presentations touting the growing popularity of eco-tourism. Bates figures that $40,000 could go a long way towards making his motel "greener" and thus more attractive to a new class of environmentally conscious guests. Bates has enlisted the services of Arborgrass Consulting to audit the Bates Motel's environmental footprint. The consulting company has identified a number of opportunities for improvement that Bates is now considering. Between managing the motel and caring for his sick mother, though, Bates has not had time to analyze the options rigorously. Please help him by answering the following questions.undal r at the Balle identified guest showers as the most significant use of water. They calculated that the lengths of guest showers at the Bates Motel are normally distributed with a mean of 10 minutes and a standard deviation of 3 minutes. a) (4 pts) What is the probability that a randomly selected shower took more than 12 minutes? b) (4 pts) Suppose on a particular Tuesday night two years ago there were eight guests checked into the hotel and that each guest took exactly one shower. What is the probability that the average length of the eight showers was more than 12 minutes? c) (4 pts) On that same Tuesday night, what is the probability that exactly four of the eight showers took longer than 12 minutes? The Arbograss report confirmed that water usage of guest showers was also normally distributed with a mean of 30 gallons and a standard deviation of 10 gallons. d) (5 pts) Suppose that Bates wants to collect his own data on water usage. He wants to compute a 95% confidence interval of the form + 1 gallon for the true mean water usage of a shower. Assuming Arborgrass' estimate of standard deviation is accurate, how many showers must Bates monitor in total to be able to compute such a confidence interval? While Bates can think of a number of creative ways to shorten guests' showers, he is leaning instead towards installing low-flow shower heads in all guest bathrooms. He installed low-flow shower heads in a few guest bathrooms as a trial, and he has so far monitored water usage from 40 showers in these rooms. He entered the number of gallons consumed in each shower into Excel and computed the following descriptive statistics. Shower | Water Usage Descriptve Statrics 23.5 18.6 Mean 24.93 26.2 Standard Error 0.92 4 31. Median 23.66 5 31.0 Mode 6 33.7 Standard Deviation 5.84 14.1 Sample Variance 34.06 23.8 Kurtosis 0.60 TO.5 Skewness 0.15 10 19.6 Range 22.80 Minimum 14.08 Maximum 36.88 Sum 997.19 40 28.8 Count 40e) (5 pts) Compute a 98% confidence interval for the true mean water usage of a shower when a low-flow shower head is installed. f) (8 pts) Arborgrass Consulting asserted that, with a low-flow shower head installed, an average shower should use approximately 24 gallons. Does Bates' data contradict Aborgrass' assertion? Perform the following hypothesis test at the 5% level of significance. Ho: U = 24 versus Ha: U # 24 Show all relevant calculations, draw a picture, and explain how you reached your conclusion. Give your conclusion in terms of the statement of the problem. What is the prob-value for the test? Because of new studies documenting toxic off-gassing from vinyl shower curtains, Arborgrass Consulting also recommended Bates consider a new line of organic shower curtains made from bamboo and hemp. Bates is intrigued, but he's worried whether he can afford it or not. Shower curtains at the Bates Motel are frequently getting ripped, and Bates incurs significant costs to replace them. g) (5 pts) Suppose that a shower curtain becomes ripped on half of the motel's guest visits. What is the probability that Bates will have to replace at least one but not more than five shower curtains tomorrow morning? There are 10 guests staying at the hotel tonight. On second thought, Bates doesn't know if "half" is correct. He looked through his records, and found that of the most recent 150 guest visits at the motel, the shower curtain has been ripped and required replacement 85 times. h) (5 pts) Obtain a 90% confidence interval for the true proportion of guest visits that result in a ripped shower curtain. (8 pts) Bates thinks that switching to organic shower curtains makes financial sense as long as shower curtains are ripped no more than 60 percent of the time. Therefore, he wants to know if his data provides compelling evidence that the true rip rate is smaller than 60 percent. He (correctly) wishes to test: Ho: It = 0.60 versus Ha: II = 0.60 Perform this hypothesis test at the 10% level. Show all relevant calculations, draw a picture, and explain how you reached your conclusion. Give your conclusion in terms of the statement of the problem. What is the prob-value for the test