Problem1: To illustrate that the mean of a random sample is an unbiased estimate ofthe population me

Problem1:

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, 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 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 ̅’sequals the population mean of12.
Problem2:
Suppose X1, X2, X3denotes a
random sample from a population with an exponentialdistribution.
a.
Show that the following are all unbiased
estimators for the populationmean.
Recall that for the exponential distribution
E(X)=1⁄

̂1 = 1

̂2 =
̂3 =

1+ 2
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2
1+ 2 2
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3

̂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).

Problem3:
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 assumethat a person is “innocent
until proven guilty.” 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?

Problem6:

Calculate
the P-value for the following hypothesis tests, based on the given valueof the teststatistic
a.
Ho:μ = μo versusH1:
μ > μo with
zo
= 1.53
b.
Ho:μ = μo versusH1: μ ≠ μo withzo = 1.95
c.
Ho:μ = μo versusH1: μ < μo=”” withzo=”−1.80″>