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, and27
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 youâve found them all â you
Calculate the mean ( ̅)for each of the samples. Assign each mean value a probability of 1/10
and verify that the mean of the ̅âsequals the population mean of12.
Suppose X1, X2, X3denotes a
random sample from a population with an exponentialdistribution.
Show that the following are all unbiased
estimators for the populationmean.
Recall that for the exponential distribution
̂1 = 1
1+ 2 2
̂4 = ̅
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).
the United States judicial system, a jury is often tasked with deciding if a
defendant is innocent or guilty. The jury is instructed to assumethat a person is âinnocent
until proven guilty.â Use this information to construct a table of the possible
outcomes of ajury 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?
the P-value for the following hypothesis tests, based on the given valueof the teststatistic
Ho:μ = μo versusH1:
μ > μo with
Ho:μ = μo versusH1: μ ≠ μo withzo = 1.95
Ho:μ = μo versusH1: μ < μo=”” withzo=”−1.80″>