STA 138 Winter 2019
Homework 01 - Due Friday, January 18th
Book Portion (does not require R)
Note: This may be hand written or typed. Answers
should be clearly marked. Please put your name in
the upper right corner.
1. Each of 40 multiple choice questions on an exam has five
possible answers, where only one is correct. A particular
student chooses the strategy of selecting an answer
randomly.
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(a) What is the distribution of total correct answers?
Specify the parameters of the distribution.
(b) What is the expected number of correct answers, and
the standard deviation of the number of correct answers?
(c) Calculate the interval that represents two standard
deviations from the mean.
(d) Would it be unusual for this student to score 50% or
more on the exam? Explain.
2. Continue with problem 1.
(a) Specify the distribution of ni
, where ni = the number
of times the student picked answer i, i = 1, 2, 3, 4, 5.
(b) Find the expected value and variance of ni
.
(c) What is the probability that the student picks each
choice exactly 8 times (8 (a)’s, 8 (b)’s, 8 (c)’s etc..).
(d) What is the correlation between n1 and n2?
3. Suppose that the probability of getting an A in a particular
course is 0.08, and assume that the all student grades
are independent. If you randomly sample 20 students
taking the course;
(a) Find the expected number of students that will get an
A, and the standard deviation of number of students
that will get an A.
(b) Find the probability that no student gets an A.
(c) Find the probability that at most 2 students get an
A.
(d) Find the probability that between 2 and 4 students
get an A (inclusive).
(e) If (in a different course), the probability that no students
out of 20 got an A was 0.1000, what was the
probability of a success? You may assume that all
students were independent, and the probability of
an A does not change.
4. In his autobiography A Sort of Life, British author Graham
Greene described a period of of severe mental depression
during which he played “Russian Roulette”.
This “game” consists of putting a bullet in one of six
chambers of a pistol, spinning the chambers to select one
at random, and then firing the bullet one at one’s head.
(a) Green played the game six times (resetting the chamber
every time) and none of them resulted in the bullet
firing. What was the probability of this outcome?
(b) What is the probability that, without resetting,
one could fire the gun 4 times in a row without the
bullet firing?
5. The scheduling manager for a certain hydro-power utility
company knows that there are an average of 12 emergency
calls regarding power failures per month. Assume
that a month consists of 30 days.
(a) Find the probability that the company will receive
exactly 10 emergency calls during a specified month.
(b) Find the probability that the company will receive
at least 1 emergency call in a given day.
(c) Suppose the utility company can handle a maximum
of 2 emergency calls per day. What is the probability
that there will be more emergency calls than the
company can handle on a given day?
(d) Find the expected number of calls per year, and the
standard deviation.
6. The marketing manager of a company has noted that she
usually receives 10 complaint calls from customers during
a week (assume a week has 7 days), and that the calls
are independent.
(a) Find the probability that she receives exactly 5 complaint
calls in one week.
(b) Find the probability that she receives at least 2 complaint
calls in one day.
(c) Find the expected number of complaint calls in one
month (assume a month has 30 days).
(d) If the rate of calls increases, would the probability in
(a) decrease or increase? Explain.
7. Suppose that a person invests in 6 stocks, each of which
has a 40% of having no return, a 40% chance of having
a positive return, and a 20% chance of having a negative
return. You may assume the stocks are independent, and
the probabilities do not change.
(a) Find the probability that 2 stocks have no return, 2
have a negative return, and 2 have a positive return.
(b) Find the probability that at least one stock has a
positive return.
(c) Find the expected value for each outcome, and the
standard deviation.
(d) What are the pairwise correlations between the different
counts (ni
’s)?
1
R Portion (requires some use of R)
Note: You do not have to use R Markdown to turn
in the homework, but the homework must be turned
in in a reasonable format. The answers to the questions
should be in the body of the homework, and
the code used to obtain those answers should be in
an appendix. There should be no code in the body of
the homework. You can accomplish this in R, Word,
LaTex, Google Docs, etc.
I. Online you will find the file “PHD.csv”. The csv file has
the following columns:
Column 1. Year: How many years it took the candidate
to graduate with a Ph.D
Column 2. Uni: Which university the subject studied at
(Berkeley, Columbia, Princeton).
Column 3. Res: Residency of subject (permanent,temporary)
Use this dataset in problems I, II, III.
Source: Espenshade, T.J. and Rodr′?guez, G. (1997).
Completing the Ph.D.: Comparative Performances of
U.S. and Foreign Students. Social Science Quarterly,
78:593-605.
(a) Find the average years to graduation for the three
schools. Which school had the highest average?
(b) Find the standard deviation of years to graduation
for the three schools. Which school had the lowest
deviation from the mean?
(c) Did temporary or permanent residents take longer
to graduate on average? Justify your answer.
(d) Find the five number summary of the number of
years it took to graduate. Do you believe the minimum
is an outlier? Justify your answer.
II. Continue with the “PHD.csv” dataset.
(a) Create a boxplot of years it took for the subjects
to graduate by residency. Do you believe there is a
significant difference between the groups? Explain.
(b) Create a histogram of years to graduate by school.
Which year had the most subjects in it for Princeton?
(c) Create a mosaic plot of residency and school.
Grouping by school, who had the highest proportion
of temporary residents?
(d) Create a mosaic plot of residency and school.
Grouping by residency, who had the highest probability
of going to Berkeley?
III. Continue with the “PHD.csv” dataset. For the following
problems, you must show results from either a plot, a table,
or an aggregate command to back up your answers.
(a) How may subjects were there attending Columbia?
(b) How different was each schools average time to graduate
compared to the overall average?
(c) Were there more people who attended Berkeley
and were temporary residents, or who attended
Columbia and were temporary residents (looking at
the absolute magnitude).
(d) If you were told a subject came from Princeton and
was a temporary resident, what would you estimate
their years to graduate to be (based on only summary
statistics and plots)? Note, if you want to
subset the data in some way but do not know how,
ask on Piazza.
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