Download Old Test 2

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STAT – Test 2: Summer 2006
NAME:______________________________
1. The manager of a computer store has kept track of the number of computers sold
per day. On the basis of this information, the manager produced the following list
of the number of daily sales
Number of Computers Sold Probability
0
.2
1
.3
2
.4
3
.1
a.
b.
c.
d.
What is the probability of selling at least one computer next Monday?
What is the probability of selling 2 “or” 3 computers next Monday?
What is the probability of selling 2 “and” 3 computers next Monday?
What is the expected number of computers sold next Monday?
2. The following table lists the joint probabilities associated with smoking and lung
disease among 60 – 65 year old men.
Is a Smoker Is a Nonsmoker
He has Lung Disease
.20
.05
He does not have Lung Disease
.24
.51
One 60 – 65 year old man is selected at random. What is the probability of the
following events?
a. He is a Smoker
b. He has lung disease
c. He has lung disease given he is a smoker
d. He is a Smoker “and” has lung disease
e. He is a Smoker “or” has lung disease
3. Transplant operations for hearts have the risk that the body may reject the organ.
A new test has been developed to detect early warning signs that the body may be
rejecting the heart. However, the test is not perfect. When the test is conducted
on someone whose heart will be rejected, approximately two out of ten tests will
be negative (the test is wrong). When the test is conducted on a person whose
heart will not be rejected, 10% will show a positive test result (another incorrect
result). Doctors know that in about 50% of heart transplants the body tries to
reject the organ.
a. Suppose the test was performed on my mother and the test is positive
(indicating early warning signs of rejection). What is the probability that
the body is attempting to reject the heart?
b. Suppose the test was performed on my mother and the test is negative
(indicating early warning signs of rejection). What is the probability that
the body is attempting to reject the heart?
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4. What are the possible values for the following random variables.
a. The number of accidents that occur on a busy stretch of highway.
b. The amount of money students earn on their summer job.
c. The time it takes your professor to drive home.
5. You first roll a die (cubic thing with numbers 1,2,3,4,5 and 6 on each face) and
then flip a coin (resulting in a head or tail).
a. Draw a probability tree to describe this experiment.
b. Calculate the probability of getting a “3” on the die and a “head” on the
coin.
6. The mean grade on the first test in this class was 85 and standard deviation 5. If
your professor decided to “curve” the grades by adding “5” to each student’s
grade,
a. what would the mean grade now be?
b. What would the standard deviation of grades now be?
7. Given a binomial random variable with n = 15 and p = .25, find the following
probabilities
a. P(X = 5) =
b. P( X < 5) =
c. P(3 < X < 5) =
8. When people sky dive for the first time, about 20% get sick. If 11 people go up in
a plane to sky dive, what is the probability that exactly 2 get sick. List and
explain any assumptions you make.
9. A continuous random variable X is uniformly distributed between 0 and 1.
a. Draw the density function
b. What is the density function (the equation)
c. Find P(X > 0.8) =
d. Find P(X > 0.8) =
10. Find the following probabilities (for the standard normal distribution with mean 0
and standard deviation 1.0). Draw a picture of the normal distribution and show
(shade) the area that represents the probability you are calculating.
a. P( Z < 1.72) =
b. P(Z > 1.72) =
c. P(Z < -1.72) =
11. The time (Y) it takes your professor to drive home each night is normally
distributed with mean 15 minutes and standard deviation 2 minutes. Find the
following probabilities. Draw a picture of the normal distribution and show
(shade) the area that represents the probability you are calculating.
a. P(Y > 20) =
b. P(Y > 25) =
c. P( 11 < Y < 19) =
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d. P (Y < 18) =
e. P( 17 < Y < 19) =
12. The manufacturing process used to make “heart pills” is known to have a standard
deviation of 0.1 mg. of active ingredient. Doctors tell us that a patient who takes
a pill with over 6 mg. of active ingredient may experience kidney problems. Since
you want to protect against this (and most likely lawyers), you are asked to
determine the “target” for the mean amount of active ingredient in each pill such
that the probability of a pill containing over 6 mg. is 0.0035 ( 0.35% ). You may
assume that the amount of active ingredient in a pill is normally distributed.
a. Solve for the target value for the mean.
b. Draw a picture of the normal distribution you came up with and show the
3 sigma limits.
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13. The amount of active ingredient in blood pressure medication (pills) is known to
have a mean of 20 units and standard deviation of 2 units.
a. If you decide to sample 400 pills and calculate the sample mean, what is
the sampling distribution of the sample mean. List any assumptions you
make.
b. If you decide to sample 4 pills and calculate the sample mean, what is the
sampling distribution of the sample mean. List any assumptions you
make.