Question 4Let x₁, x₂, …xₓ be a random sample of n i.i.d. observationsfollowing the Poisson distribution with parameter (mean) ϴ such that E(Xₐ) = ϴ. Consider three estimators of ϴ given as:ϴ₁ = X, ϴ₂ = X₁, ϴ₃ = (x₁ + 2x₂)/3i.) Show that all the three estimators are unbiasedii.) Calculate which of three has the least VarianceQuestion 5A psychology researcher wants to find out if exercisingbefore taking a quiz affects a student’s performance. To test this he randomly assigns students toeither exercise for 10 minutes before taking a short quiz, or to take the quizwithout exercising first. 37 studentsexercise first and averaged 84% on the quiz with a standard deviation of 7%while 32 students skip exercising and score 81% with a standard deviation of6%. Neither sample was substantiallyskewed.a.) Write down the null and the alternativehypothesis in the test.b.) Draw a picture of the samlpling distributionassuming that the null hypothesis is true, shading in the regions(s) that wouldresult in a type 1 error (falsely rejecting the null hypothesis assuming thatit’s true of 0.05c.) Test the hypothesis at 95% confidence. NOTE: Instead of assuming that the variances are equal, first test for theequality of variances. Once you concluderegarding the equality of variances, test for the equality of mean performance.d.) What is the P-value of the test? Question 6a.) For the IND ENG 716 group project, members ofgroup 5 needed to determine if the percentage titanium content std. dev. In an aerospacecasting is within the acceptable specifications of SD = .30. To do this, they tested the percentagecontent of titanium in 51 casting specimen and the sample std. dev. was 0.37. Construct the 95% confidence interval of thestandard deviation and comment on whether the company’s castings are within theallowable specificationsb.) Next the group needed to study if the two allyimpurity level detection tests that the company uses perform the same. To do this, they tested eight specimens onboth apparatus and recorded the following impurity levels: Is there evidence at 90% confidence thatthe tests differ in mean impurity levels? Hint: be careful to decide ifthese are homogenous samples or not, hence use the appropriate test. What is the P-value of your test?SpecimenTest 1Test 211.22.021.31.731.51.541.41.351.72.061.82.171.41.781.01.6

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