A coin that lands heads 25%, 50% or 75% of the time. Before seeing any
data you assume that each possibility is equally likely. You toss the coin 10
times, and it lands heads 7 times. (20%)
(a) What is your posterior probability that p = .5 ?
(b) What is your prediction for the probability that the coin will land heads
the next time it is tossed?
(c) Suppose you wanted to test H0 : p = .5 against H1 : p > .5 . What is
the observed p-value?
There are 20 black balls and 5 red in a box. Each time one ball is chosen at
random (with replacement). (15%)
(a) What is the chance that a black ball is chosen on the first three draws?
(b) What is the chance of choosing exactly two red balls in the first five
draws?
(c) What is the chance of choosing a red ball on the fifth draw given that
you chose a black one on each of the first four draws?
An important part of the customer service responsibilities of a natural gas
utility company concerns the speed with which calls relating to no heat in
a house can be serviced. Suppose that one service variable of importance
refers to whether or not the repair person reaches the home within a
two-hour period. Past data indicate that period. If a sample of five service
calls for “no heat” is selected, what is the probability that the repair person
will arrive at
(a) At least three houses within the two-hour period? (5%)
(b) In this typical textbook exercise, what questions would you like to ask
if you were the manager of a customer service center? Explain why.
(10%)
The director of a large employment agency wishes to study various
characteristics of its job applications. A sample of 200 applicants has been
selected for analysis. Seventy applicants have had their current jobs for at
least five years; 80 of the applicants are college graduates; 25 of the college
graduates have had their current jobs at least five years.
(a) What is the probability that an applicant chosen at random
(1) Is a college graduate and has held the current job less than five
years? (3%)
(2) Is a college graduate or has held the current job at least five years?
(3%)
(b) Given that a particular employee is a college graduate, what is the
probability that he or she has held the current job less than five years?
(5%)
(c) Determine whether being a college graduate and holding the current
job for at least five years are statistically independent. (4%)
(d) As the director, can you explain the managerial meanings and
implications from the answer of (b)? (10%)
There are two different kinds of estimators: Maximum Likelihood
Estimator (MLE) and Least Squares Estimator (LSE). Why do we
“maximize” the likelihood function to get the MLE? And why do we need to
“minimize” the loss function to get the LSE? Are they always the same?
Give an example to explain. (10%)
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