Download 1. True or False - Circle the correct response. (1 point each)

Statistics 224/324, Spring 2007
Midterm Exam 1: Wednesday, February 28th, 2007
* Remember to write your name below, on the tops of pages 1 and 2, and on the blue book.
* With the exception of the True/False and Multiple Choice, write all answers and work in the
blue books.
* Show all work to get full credit – partial credit will be awarded.
* When you are finished, tuck your exam into your blue book.
* Use your knowledge and skills to the best of your ability!
NAME_______________________
Your Points
True or False
Possible Points
6
Multiple Choice
14
Fish! French Fish!
15
Throwing Dice Along the Wharf
17
Sledding
9
So Hot, You Could Cook a Hog’s Rear
in a Spoonful
TOTAL
14
75
Name_______________________
1. True or False - Circle the correct response. (1 point each)
a. For two events A and B in the same sample space S, if A is independent of B, then A is
also independent of BC. TRUE
FALSE
b. For a continuous random variable Y with pdf f(y), P(Y = y) = f(y).
TRUE
FALSE
c. An event A and its complement AC are independent of one another.
TRUE
FALSE
d. Statistics uses a sample to make inferences about a population
TRUE
FALSE
e. A random variable that takes value 1 with probability 0.3 and value 3 with probability 0.7
is a Bernoulli random variable.
TRUE FALSE
f. The sample standard deviation is a measure of the center of a set of data.
TRUE FALSE
2. Multiple Choice – Circle the correct response. (2 points each)
A. If X ~ Bin(14, 0.6), then P(X = 5) is:
a. 0.0000
b. 0.0003
c. 0.0408
d. 0.0778
e. 0.2066
B. How many ways are there to deal a hand of 13 cards out of a deck of 52 different
cards, if order is unimportant? (This is the number of possible bridge hands.)
a. 6,227,020,800
= 13!
b. 6.350135596 * 1011 = 52 choose 13
c. 3.954242644 * 1021 = 52! / 39!
d. 8.065817517 * 1067 = 52!
e. None of these
Name_______________________
C. If X ~ Exponential(3), then P( X > 2) is:
a. 0.0025
b. 0.0074
c. 0.1991
d. 0.9926
e. 0.9975
D. For events A and B in the same sample space S, if P(A) = 0.4, P(B) = 0.2, and
P(A|B) = 0.5, then P(BC| A ) is:
a. 0.10
b. 0.25
c. 0.50
d. 0.75
e. 0.90
E. If U ~ Unif(0,4), then P( -1 < U < 2.5) is:
a. 0.375
b. 0.400
c. 0.625
d. 0.875
e. None of these
F. If X and Y are independent random variables with VAR(X) = σ2 and VAR(Y) = τ2,
then VAR [(X-Y)/3] is:
a.
b.
c.
d.
e.
(σ2 - τ2) / 3
(σ2 + τ2) / 3
(σ2 - τ2) / 9
(σ2 + τ2) / 9
None of these
G. If X ~ Bin(1200, 0.002), use the Poisson approximation to the binomial to calculate
P(X = 3).
a. 0.013115
b. 0.209014
c. 0.209244
d. Poisson approximation to the Binomial should not be used here
e. None of these
3. Fish! French Fish! (15 total points)
The following data represent the number of fish caught off the southern coast of France on 16
consecutive days (sorted for your convenience):
5, 9, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 18, 19, 20
a. Create a stem and leaf plot for these data. (3 points)
b. Find the sample median, IQR, sample mean, and sample variance of these data. You may
use the fact that Σ(xi2) = 3087. (8 points)
c. Is it possible that these data were generated as a Poisson process? Justify your answer by
determining if these data are consistent with the properties of a Poisson RV. (4 points)
4. Throwing Dice Along the Wharf (17 total points)
In my new riverboat casino, I offer two dice games:
GAME 1: You are allowed 3 rolls of a fair, six-sided die, numbered 1 through 6. You bet $1,
and if you can get at least one 6 in those three rolls, I pay you $2 (and you make a net profit of
$1).
GAME 2: You again bet $1. In the first round, you roll one fair, six-sided die, numbered 1
through 6. If you get a 6 on the first roll, you win $4 (net profit of $3), and you stop. If you get
a 1, 2, or 3 on the first roll, you get nothing, and you stop. If you get a 4 or 5 on the first roll, I
give you two dice to roll on the second round. If at least one die thrown in the second round
shows 6, you win $1 (net profit of $0).
a. For each of the games above, define a RV that represents your profit as a player (if you
lose money, denote that as negative profit). Compute the pmf of each of these RVs. (13
points)
b. Compute the expected value of each of the RVs defined in (a). (4 points)
5. Sledding (9 total points)
At a certain sledding hill, there are three types of sleds – sno-tubes, saucers, and toboggans.
When you get to the top of the hill, the attendant randomly selects a type of sled and pushes you
on your way. There are two possible outcomes of a trip down the hill – either you survive to the
bottom, or you wipe-out. Here are some relevant probabilities:
P(sno-tube) = 0.1
P(saucer) = 0.4
P(survive | toboggan) = 0.3
P(wipe-out | saucer) = 0.8
P(survive | sno-tube) = 0.7
a. What is the probability of surviving to the bottom? (4points)
b. If you see somebody in line who is full of snow (they wiped-out) what is the probability
that they rode a toboggan on the previous trip down the hill? In other words, find
P(toboggan | wipe-out). (5 points)
6. So Hot, You Could Cook a Hog’s Rear in a Spoonful (14 total points)
A famous chili chef has a very unpredictable hot sauce dispenser. Each dispensation releases a
random amount of sauce, where the amount is approximately distributed as a normal RV. The
chef knows that anywhere between 2.82 and 5.18 grams of hot sauce will make an acceptable
batch of chili.
a. Initially, the amount dispensed is normal with mean 5g, and variance 2g2, i.e., if X is the
amount dispensed, X ~ N(5,2). What is the probability of a good batch of chili? [i.e., find
P(2.82 < X < 5.18) ]. (5 points)
b. The chef discovers that he can change the mean amount dispensed by twisting an
adjustment screw on the back of the dispenser; the variance, however, remains the same
regardless of the mean. To what value should he set the mean so that the probability of a
good batch of chili is maximized? [i.e., find the value of μ such that P(2.82 < X < 5.18) is
maximized.] (3 points)
c. With some further searching, the chef discovers yet another screw which adjusts the
variance. Using the mean as calculated from (b), determine to what value the chef should
set the variance so that the probability of a good batch of chili is 0.95. [If you cannot find
the answer to (b), use ‘μ’ everywhere you would use the mean, and simplify the equation
as much as possible, for full credit.] (6 points)