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ENGRD 2700: Basic Engineering Probability and Statistics Fall 2023 Homework 7 Due Wednesday, November 8th at 11:59pm. Submit Solutions to Gradescope by clicking the name of the assignment. See syllabus for detailed submission instructions. When completing this assignment (and all subsequent ones), keep in mind the following: • You must complete the homework individually and independently. • Provide evidence for each of your answers. If a calculation involves only very minor computation then explain the computation you performed and give the results. If a calculation involves more complicated steps on many many records then hand in the calculations and formulas for the first few records only. • Write clearly and legibly. You are encouraged to type your work although you do not have to. We may deduct points if your answers are di ffi cult to read or disorganized. • For questions that you answer using Python, attach any code that you write, along with the relevant plots. • Submit your homework a single pdf file on Gradescope.

1. A sample of 100 service times at a call center has a sample mean of 9 minutes and a sample standard deviation of 6 minutes. Assume that the service times are independent and have a normal distribution. (a) Give a 95% confidence interval for the mean service time. (b) Approximately how many service times we would have to collect to return a 95% confidence interval whose width is at most 15 seconds (= 1/4 minutes)? (c) We want to quote a time T to a customer so that, with 95% probability, the (random) service time will be less than T . Does your answer in Part (a) help? If yes, how? If no, explain why not.

2. Ratan Tata and Pete Coors are running for state governor, and two well known Super-PACs, Sega- Genesis and Super-Nintendo, decide to gauge public opinion. (a) Super-Sega-Genesis interviews 453 people, and finds that 55% of individuals want to vote for Ratan. Construct a 95% confidence interval for p , the proportion of all voters in the state sup- porting Ratan. (b) Super-Nintendo conducts its own independent study, and obtains the interval [0 . 492 , 0 . 568] from a sample size of 378. What confidence level did Super-Nintendo use? (c) Suppose, contrary to the information specified in part (b), that Super-Nintendo obtained the 90% confidence interval [0 . 509 , 0 . 591] instead. How many individuals did Super-Nintendo interview?

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