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## Problem 1:To illustrate that the mean of a random sample is an unbiased estimateof the population me

Problem 1:To illustrate that the mean of a random sample is an unbiased estimateof the population mean, consider five slips of paper numbered 3, 6, 9,15, and 27a. List all possible combinations of sample size 3 that could be chosenwithout replacement from this finite population (you can use thecombination formula to make sure youâve found them all â you shouldhave 10)b. Calculate the mean (??) for each of the samples. Assign each mean?value a probability of 1/10 and verify that the mean of the ??âs equals?the population mean of 12.Problem 2:Suppose X1, X2, X3 denotes a random sample from a population with anexponential distribution.a. Show that the following are all unbiased estimators for thepopulation mean. Recall that for the exponential distributionE(X)=1????1 = ?1??2 = ?1+ ?22??3 = ?1+2 22?3??4 = ??b. How would you determine which of the unbiasedestimators above is the most efficient? (You do notneed to do any calculations, just provide anexplanation).Problem 3:In the United States judicial system, a jury is often tasked with deciding if adefendant is innocent or guilty. The jury is instructed to assume that aperson is âinnocent until proven guilty.â Use this information to construct atable of the possible outcomes of a jury trial, in terms of the actual guilt orinnocence of the defendant and the jury verdict. In this context, whatsituation results in a Type I error? What about a Type II error?Problem 6:Calculate the P-value for the following hypothesis tests, based on the givenvalue of the test statistica. Ho: ? = ?o versus H1: ? > ?o with zo = 1.53b. Ho: ? = ?o versus H1: ? ? ?o with zo = 1.95c. Ho: ? = ?o versus H1: ?