Download Sample exam 2

Name___________________________
I certify by signing that I have neither
given, received, nor witnessed any
unauthorized assistance on this exam.
Exam 2---Math 2210---Fall 2013
Directions: Follow all directions. Show all work where necessary (excluding computations done
exclusively on the calculator) neatly in detail to ensure proper credit. Problems with no work where
work is indicated may receive no credit.
1. Suppose you must take two exams today. First, you’ll take a Stats exam with Dr. Smith, then
you’ll take a Philosophy exam with Dr. Weiss. If you get an A on Smith’s exam, this might be a sign
that you did a good job preparing for your exams, and there is a 80% probability you’ll get an A on
Weiss’ exam. If you don’t get an A on Smith’s exam, then maybe you didn’t prepare so well and
there will be only a 10% chance that you’ll get an A on Weiss’ exam. You think initially there is a
40% chance you’ll get an A on Smith’s exam. Dr. Weiss actually hands back his exam first, and you
find that you got an A on Weiss’ exam. Use Bayes’ Rule to find the probability P(S|W) that you got
an A on Smith’s exam given that you know you got an A on Weiss’ exam.
2. Answer each question by underlining the appropriate answer.
In parts a and b, determine if the random variable under consideration is discrete or
continuous.
a. Let X = the number of heads obtained when tossing a coin 75 times.
CONTINUOUS
DISCRETE
b. Let X = the total time in minutes it takes you to complete problem 6 on this exam.
CONTINUOUS
DISCRETE
In parts c and d, determine whether or not the random variable in question has a binomial
distribution.
c. An officer from public safety sees 100 cars in the Allgood parking lot. The officer proceeds to run
the plate number of each car and sees how many outstanding parking tickets are assigned to each car.
BINOMIAL
NOT BINOMIAL
d. An officer from public safety sees 100 cars in the Allgood parking lot. The officer proceeds to
inspect each car and see if it has a valid parking tag, and then tallies up the number of cars without a
valid tag.
BINOMIAL
NOT BINOMIAL
3. You believe the vending machine in Allgood Hall will eat (accept but not credit or refund) any
given dollar bill you try to use with probability 1.5%. Over the course of one year, you try to use 100
dollars in the machine. Assume each bill used is independent.
a. If X = the mean number of your dollars eaten by the machine, what is the mean and standard
deviation of the random variable X?
b. Find the probability the machine eats 6 or more of your dollars.
c. Suppose you do in fact try to use 100 dollars in the machine over the course of the academic year,
and the machine eats 6 of them. Which of the following would you conclude? Underline your
answer of choice.
Assuming the machine eats 1.5% of the bills you try to use, seeing the machine eat 6 or more
of 100 bills is not particularly out of the ordinary.
You might suspect that perhaps the machine is in fact eating more than 1.5% of the bills
used.
4. According to the Augusta Chronicle, there were 32 homicides in Richmond County during 2012.
Assuming the times between each homicide are independent of one another, the number of
homicides per unit time should follow a Poisson distribution.
a. Find the probability there would be at least one (i.e. one or more) homicides on any given day in
Richmond County.
b. Find the probability there would be exactly one homicide during a given five-day period.
5. You believe that if you attempt to park your car on campus without displaying a parking tag, there
is a 70% probability you will be ticketed by Public Safety.
a. If you park your car on campus 10 times without a tag, find the probability you’ll get ticketed all
10 times. Assume independence.
b. Find the probability you get ticketed exactly 5 times.
6. In Denmark, it is estimated that a given person will eventually develop brain cancer with
probability .000340. In a study of 420,095 cell phone users in Denmark, it was found that 135 of
them developed brain cancer.
a. Let X = the number of people in a sample of size 420,095 who eventually develop brain cancer.
Find the mean and standard deviation of X.
b. Find the probability that 135 or more people in a sample of this size would have eventually
developed brain cancer.
7. The lengths of pregnancies (which end with live births) in humans are normally distributed with
mean 268 days and standard deviation 15 days.
a. Find the probability a human pregnancy would last 308 or more days.
b. Suppose any pregnancy whose duration is in the bottom 4% of the distribution is deemed
premature (as the child born might need special care in such a scenario). What pregnancy length (or
shorter!) would be classified as premature?
8. Suppose that when you are exposed to a certain contagious disease (say the common cold, being
in close proximity to a person when they sneeze), there is a 3% probability that you will become
infected with the disease.
a. If you are exposed to the disease X times, write a formula for the probability that you get infected
at least once (i.e. one or more times).
b. Hopefully using the calculator, what is the minimum number of times must you be exposed to
have a 90% probability of getting infected (one or more times)? You can save yourself some
needless screen-scrolling if you pick a judicious starting value.