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Question 1:
The even numbers of a die are painted red, the odd numbers white. The die is rolled once.
Define the event A as ‘2 or 3’, and the event B as ‘a red face’. Are these events independent?
Explain.
Question 2:
In the world cup of 2014, four teams make it to the semifinals: Brazil, Cameroon, Turkey, and
Germany. Turkey will play Brazil, and can win with probability 0.35. Germany will play
Cameroon, and can win with probability 0.6. If the Turkish team makes it to the final, they
can beat Germany, or Cameroon, in the final with probability 0.55, or 0.65, respectively.
Brazil can beat Germany, or Cameroon, in the final with probability 0.6, or 0.7, respectively.
a) What is the probability that the final is between Turkey and Germany?
b) What is the probability that Turkey will win the world cup ?
c) Given that Brazil won the world cup, what is the probability that Cameroon beat Germany?
Question 3:
There are 37 numbers in a data set. You want to find the inter-quartile range (IQR). What do
you do? Give your answer in as much detail as possible. (Detail does NOT mean a lot of
words – I will take off points for irrelevant information)
Question 4:
There are three -3’s, four -1’s, two 0’s, two 1’s, four 2’s, and one 3 in a data set.
a) Draw a relative frequency histogram.
b) Find the mode, mean, median, and 70th percentile. Which of these will change if we add
two -3’s and two 3’s? Which will not change? Why?
c) Find the variance and standard deviation.
Question 5:
A bivariate numerical data set consists of the pairs of values (-2,-3), (-1,-3), (0,-2), (1,-2), (2,1), (3,0), (4,1) for the variables (x,y).
a) Find the correlation coefficient
b) Find the least square fitted line
c) What is the vertical error E3 for the third point (0,-2) ?
Question 6:
At the gamblers buffet there are six identical covered containers of food, four containing one
kind of food. Two are empty. You are asked to pick two containers at random. After picking
two containers, you may remove the lid and take as much food from these two as you want.
What is the probability that you will eat only one kind of food?
Question 1:
Histogram of X
16
Mean
StDev
N
14
13.08
1.608
60
Frequency
12
10
8
6
4
2
0
10
11
12
14
13
15
16
17
X
Sum = 785
Sum of squares = 10423
a) What is the number of data points?
b) What is the mode and the median?
c) What is the relative frequency of 12?
d) The 100pth percentile is 14.5. What is p? (Expressed as a rational number)
e) What is the standart deviation? What is the sum of squares of deviations?
f) Make a stem-and-leaf display for the above data.
(2)
(5)
(2)
(5)
(6)
(4)
Question 2:
A bag has 3 blue and 8 green balls. Two balls are drawn in succession without replacement
from the bag. Let X be the random variable representing the number of green balls drawn.
a) What is the probability that X=2?
(6)
b) What is the expected value of X?
(8)
c) What is the probability that the first draw is blue, given that the second draw is green? (8)
Question 3 :
(24)
Consider the line going through the points (1,3) and (3,1). Is this the regression line for the
points P1(1,3.5), P2(2,1.5), and P3(3,1.5) ? Compute the sum of squares of vertical errors for
this line. Also, if this line is not the regression line, find it.
Question 2: (10)
According to a survey conducted by ABC Research , 60% of all consumers have called retail
companies’ call centers for information about some product before buying it. Suppose a
random sample of 20 consumers is contacted and interviewed about their buying habits. What
is the probability that 15 or more of these consumers have called a call center number for
information about some product? What is the probability that exactly 12 of these consumers
have called a call center number for information about some product?
Question 3 : (15)
Suppose 40% of all college students have a computer at home and a sample of 64 is taken.
What is the approximate probability that more than 30 of those in the sample have a computer
at home? Write an expression for the exact probability.
Question 4 : (10)
A student randomly guesses the answers to a five question true/false test. If there is a 50%
chance of guessing correctly on each question, what is the probability that the student misses
exactly 2 questions? What is the probability that the student misses no questions? (DON’T
use a table or an approximation. Find the exact answer with the aid of a calculator)
Question 5 : (40)
The sales manager at Moss Point Metropolitan Motors compiled the following discrete
probability distribution. In this distribution X represents the number of cars sold per day at
her dealership.
X
0
1
2
3
P(X)
0.25
0.35
0.25
0.15
a) Find the mean, and standard deviation of X.
b) What is the probability that more than 28 cars will be sold in the next 10 working days?
d) What is the probablity that on more than 17 days less than two cars will be sold in the next
25 working days?
Question 1:
A bad form of cancer occurs in 3% of all Aleuts. A screening test has been developed that is
good, but not perfect. The test will label 5% of healthy Aleuts as having cancer, and will label
2% of Aleuts with cancer as not having cancer. What percentage of the people who are
labeled as having cancer by the test, actually do have cancer?
Question 2:
Two dice are thrown. One is red, and one is white. The following events are defined:
E = ‘Red die is 5’
F = ‘Total is 7’
G = ‘Total is 10’
a) Show that P(F/E) = P(F)
b) Show that P(G/E)  P(G)
c) If you are told that the red die shows 5, do you want to bet on whether the dice show a total
of 7, or 10? Why?
Question 3:
a) How many ways can you put 7 pictures in 5 frames?
b) How many ways can you put 3 pictures in 7 frames?
Note: In (a) and (b), each frame is in a separate room.
c) Two out of seven pictures are fake. If you choose four pictures at random, what is the
probability that you have only one fake picture among the four?
Question 4:
The temperature for the next 6 days is going to be 32, 29, 28, 31, 34, and 37. (All numbers are
in degrees centigrade).
a) Find the average and variance of the temperatures for the next 6 days
b) Find the average and variance of the temperatures for the next 6 days in degrees
Fahrenheit. ( [Fahrenheit] = 32 + 1.8[Celsius]. Example: [86F] = 32 + 1.8[30C] ).
Question 5:
y  0.56  1.6 x is the least square line (regression line) for the following pairs of
measurements: ( x1 , 1.6), ( x 2 ,2.4), ( x3 ,3.0), ( x 4 ,3.4), ( x5 ,4.0)
a) What is x , the average of the x-values?
b) What is S yy ?
c) Assuming that x1 = 1, x 2 = 1.5, x3 = 2, x 4 = 2.5, x5 = 3, find the correlation coefficient.
d) What is the sum of squares of vertical errors for the regression line?
Question 6:
A, B, and C are three events defined on a sample space. P(A) = 0.36, P(B) = 0.5,
P(C) = 0.50, P(A  B) = 0.18, P(A  C) = 0.11, P(B  C) = 0.20, and P(A  B  C) = 0.08,
a) Investigate the independence of A and B using the product of their probabilities.
b) Investigate the independence of A and C using P(C/A).
c) Investigate the independence of B and C using P(B/C).
d) Can you find the sum of the probabilities of all simple events outside A  B  C ?
Question 6:
Of the 120,000 nomads of Kush, 30,000 own three camels, 40,000 own two camels, 45,000
own one camel, and 5,000 do not own any camels at all. The following experiment is
conducted 500 times: 10 nomads are chosen at random, and the total amount of camels they
own is recorded. The variance of the 500 numbers obtained in these 500 experiments is
calculated. What number would be very close to this calculated variance?
Question 7:
For a Bernoulli trial with p  0.45 ,
a) calculate the probability that there will be less than 11 successes in 19 Bernoulli trials
b) calculate the probability that there will be less than 11 failures in 19 Bernoulli trials
Question 1:
y  0.56  1.6 x is the least square fitted line (regression line) for the following pairs of
measurements: ( x1 , 1.6), ( x 2 ,2.4), ( x3 ,3.0), ( x 4 ,3.4), ( x5 ,4.0)
a) What is x , the average of the x-values? (You cannot use the x-values in part (c))
b) What is S yy ?
c) Assuming that x1 = 1, x 2 = 1.5, x3 = 2, x 4 = 2.5, x5 = 3, find the correlation coefficient.
d) Using the x-values given in part (c), what is the sum of squares of vertical errors for the
regression line?
Question 4:
a) The answer to the question “What is the probability that in a random draw of five cards
from a deck, all of them are spades?” was given in the lecture as C13,5 / C 52,5 . A student
objected, and said that the probability of drawing the first spade is
12
51
13
52
13 12 11 10 9
52 51 50 49 48
, drawing a spade again
is , and so forth, so that the probability of drawing five spades is
.
What do you say?
b) Two out of seven pictures are fake. If you choose four pictures at random, what is the
probability that you have only one fake picture among the four?
Question 5:
A bag has one white, two blue, and three red balls. Two balls are drawn in succession, at
random, without replacement.
a) Find the sample space and assign probabilities to the simple events
b) Find P ( B / A) , where A = ‘red on second draw’ and B = ‘blue on first draw’
c) Are A and B independent? You must use P ( B / A) to answer this question.
Question 2:
The more alcohol there is in a person’s bloodstream, the slower is that person’s reaction time
to a given stimulus. To test this, 4 volunteers were used to obtain the following bivariate data,
where the variable x denotes the amount of alcohol in a volunteer’s bloodstream, and the
variable y the reaction time to a stimulus: (0.08, 0.32), (0.12, 0.44), (0.15, 0.47), (0.18, 0.63).
Based on this, predict the reaction time for an individual with blood alcohol content of 0.20.
(Work carefully, no partial credit is given in this question)