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 you’ve found them all – 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 ??’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 “innocent until proven guilty.” 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: ?