(1) (modified from Stock and Watson Exercise 4.5) A professor decides to run an experiment to measure the effect of time pressure on final exam scores. She gives each of the 500 students in her course the same final exam, but some students have 45 minutes to complete the exam while others have 60 minutes. Each student is randomly assigned one of the examination times based on the flip of a coin. Let Y, denote the number of points (out of 100) scored on the exam by the ith student (i = 1, ..., 500), and let X¿ be the amount of minutes that the student had to complete the exam (so X₁ = 45 or X₂ = 60). Consider the linear regression model Y = Bo + BıXi+uj . (a) Explain what the term u, represents here. Why will different students have different values of ui? (b) Is it plausible to assume that E(u₁|X;) briefly. = 0 in this setting? Explain (c) Suppose the estimated regression line is Ỹ; = 52+0.4X₁. Compute the predicted score of a student that was given 45 minutes to complete the exam. Repeat for 60 minutes. (d) Based on the estimates in part (c), compute the estimated gain in score for a student who is given an additional 10 minutes on the exam. (e) Suppose the university administration would consider a policy change that requires all exams to take 120 minutes. What would the estimates in (c) literally suggest happens to students' average scores in this case?

In a science fair project, Emily conducted an experiment in which she tested professional touch therapists to see if they could sense her energy field. She flipped a coin to select either her right hand or her left hand, and then she asked the therapists to identify the selected hand by placing their hand just under Emily's hand without seeing it and without touching it. Among 340 trials, the touch therapists were correct 162 times. Complete parts (a) through (d). a. Given that Emily used a coin toss to select either her right hand or her left hand, what proportion of correct responses would be expected if the touch therapists made random guesses? 0.5 (Type an integer or a decimal. Do not round.) b. Using Emily's sample results, what is the best point estimate of the therapists' success rate? (Round to three decimal places as needed.)

In a science fair​ project, Emily conducted an experiment in which she tested professional touch therapists to see if they could sense her energy field. She flipped a coin to select either her right hand or her left​ hand, and then she asked the therapists to identify the selected hand by placing their hand just under​ Emily's hand without seeing it and without touching it. Among 331 ​trials, the touch therapists were correct 159 times. Complete parts​ (a) through​ (d). a. Given that Emily used a coin toss to select either her right hand or her left​ hand, what proportion of correct responses would be expected if the touch therapists made random​ guesses? ​(Type an integer or decial. Do not round)

In a science fair​ project, Emily conducted an experiment in which she tested professional touch therapists to see if they could sense her energy field. She flipped a coin to select either her right hand or her left​ hand, and then she asked the therapists to identify the selected hand by placing their hand just under​ Emily's hand without seeing it and without touching it. Among 354 ​trials, the touch therapists were correct 170 times. Complete parts​ (a) through​ (d). a. Given that Emily used a coin toss to select either her right hand or her left​ hand, what proportion of correct responses would be expected if the touch therapists made random​ guesses?     nothing ​(Type an integer or a decimal. Do not​ round.) b. Using​ Emily's sample​ results, what is the best point estimate of the​ therapists' success​ rate?     nothing ​(Round to three decimal places as​ needed.) c. Using​ Emily's sample​ results, construct a 95​% confidence interval estimate of the proportion of correct…