# To illustrate that the mean of a random sample is an unbiased estimate

August 30, 2017

Question
Problem 1:
To illustrate that the mean of a random sample is an unbiased estimate
of the 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
have 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 an
exponential distribution.
a. Show that the following are all unbiased estimators for the
population mean. Recall that for the exponential distribution
E(X)=1??

??1 = ?1
??2 = ?1+ ?2
2
??3 = ?1+2 2
2?
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 an
explanation).

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 II error?

Problem 6:
Calculate the P-value for the following hypothesis tests, based on the given
value of the test statistic
a. Ho: ? = ?o versus H1: ? > ?o with zo = 1.53
b. Ho: ? = ?o versus H1: ? ? ?o with zo = 1.95
c. Ho: ? = ?o versus H1: ? < ?o with zo = ?1.80

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