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 youve 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 thesequals 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
.0/msohtmlclip1/01/clip_image001.gif”>
2
1+ 22
.0/msohtmlclip1/01/clip_image001.gif”>
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: = o versusH1:
> o with
zo
= 1.53
b.
Ho: = o versusH1: o withzo = 1.95
c.
Ho: = o versusH1: < o withzo = 1.80 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, and27a. 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 youve 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 thesequals the population mean of12.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 =12 =3 =1+ 2.0/msohtmlclip1/01/clip_image001.gif">21+ 22.0/msohtmlclip1/01/clip_image001.gif”>34 =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).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?Calculate
the P-value for the following hypothesis tests, based on the given valueof the teststatistica.
Ho: = o versusH1:
> o with
zo
= 1.53b.
Ho: = o versusH1: o withzo = 1.95c.
Ho: = o versusH1: < o withzo = 1.80