1COR-STAT 1203 Introduction to Statistical Theory, G3-G4, Term II, 2021-22Assignment 4Due date and time:Friday 11.59pm,Week 11(March 25, 2022)Place to submit:eLearn→Assignments→Assignment 4Instructions:(i)Write yournameandgroup numberclearly on the top of the first page;(ii)Convert your solutions into one pdf file named:YourName_A4_GroupNo.pdf,e.g., GohZhiHui_A4_G3.pdf, TayAlbert_A4_G4.pdf;(iii) Late assignment will not be accepted, andzero markwill be given to two or morevery similar assignments;(iv) You may do the assignment by handwriting onA4-size paper→scan→pdf;or typing by word (or some other software, with A4 page specification)→pdf;(v) Minimize the size of the pdf file for submission.1.A random sampleX1,X2, …,Xnof sizenis taken from an exponential distributionwith meanβ.LetXbe the sample mean.(a)Show that𝑋𝑋�is an unbiased estimator ofβ(b)Show that Var(𝑋𝑋�) =𝛽𝛽2𝑛𝑛⁄(c)Based on a random sample of sizen= 50,𝑋𝑋�is observed to be 12.Give anapproximate 95% confidence interval forβ2.The heights of a random sample of 50 college students showed a mean of 174.5centimeters and a standard deviation of 6.9 centimeters.(a)Construct a 95% confidence interval (C.I.) for the mean height of all collegestudents.(b)What can we assert with 95% confidence about the possible size of our errorif we estimate the mean height of all college students to be 174.5 centimeters?(c)Construct a 99% C.I. for the mean height of all college students. Comparewith the 95% C.I. obtained in (a) and comment.3.The following measurements were recorded for the drying time, in hours, of acertain brand of latex paint:3.4, 2.5, 4.8, 2.9, 3.6, 2.8, 3.3, 5.6, 3.7, 2.8, 4.4, 4.0, 5.2, 3.0, 4.8.(a)Provide point estimates for the population mean and standard deviation of thedrying time.(b)Assuming the measurements represent a random sample from a normalpopulation, find a 90% confidence interval and a 95% confidence interval forthe population mean drying time.(c)Suppose that it is known (from previous studies) that the population standarddeviation of drying timeσ= 1. How large a sample will be needed in order forthe sample mean to be within 0.2 hour of the true mean with 95% confidence?...