Problem 1:
To illustrate that the mean of a random sample is an unbiased estimate ofthe population mean, consider five slips of paper numbered 3, 6, 9, 15, and 27
a. List all possible combinations of sample size 3 that could be chosen without replacement from this finite population (you can use the combination formula to make sure you've found them all - you should have10)
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 of12.
Problem 2:
Suppose X1, X2, X3 denotes a random sample from a population with an exponential distribution.
a. Show that the following are all unbiased estimators for the populationmean. Recall that for the exponential distributionE(X)=1⁄??
b. How would you determine which of the unbiased estimators above is the most efficient? (You do not need to do any calculations, just provide anexplanation).
Problem 3:
In the United States judicial system, a jury is often tasked with deciding if a defendant is innocent or guilty. The jury is instructed to assume that a person is "innocent until proven guilty." Use this information to construct a table of the possible outcomes of a jury trial, in terms of the actual guilt or innocence of the defendant and the jury verdict. In this context, what situation results in a Type I error? What about a Type IIerror?
Problem 4:
Calculate the P-value for the following hypothesis tests, based on the given valueof the teststatistic
a. Ho: μ = μo versus H1: μ > μo with zo = 1.53
b. Ho: μ = μo versus H1: μ ≠ μo with zo = 1.95
c. Ho: μ = μo versus H1: μ < μo with zo = -1.80