Problems: Drawing Tickets from a Box Part
1
1. Suppose we have this box of tickets: 0, 2, 3, 4,
6.
 a) What is the average of the box?
b) What is the standard deviation of the box?
c) If we take a draw of 100 tickets out of the box
(replacing the ticket after each draw), we expect a sum
of _________ give or take ___________.
d) If we take a draw of 100 tickets out of the box
(replacing the ticket after each draw), we expect an
average of _________ give or take ___________.
e. If we draw 100 tickets out of the box and get a sum or
350, our chance error is _____________.
2. A box has four tickets, 1, 2, 3, and 10.
 a) What is the average of the box? ___________
b) What is the standard deviation of the box?
_____________
c) If 200 draws are made from the box, we expect to get a
total of about _________ give or take about
____________.
d) If 200 draws are made from the box, what is that
probability that the sum will be 875 or greater?
3. A student is taking a multiplechoice test with 100
questions. Each question has five options and only one of
those options is correct. A correct answer is worth five
points and a wrong answer is worth nothing. Unfortunately
for the student, the test is in Russian, and he knows no
Russian, so must guess randomly.
In terms of the box model in the text, answering a
question would be like drawing tickets from the box: 5, 0,
0, 0, 0.
 a) The average of the box is _______________
 b) The standard deviation of the box is
____________________
 c) Since each correct answer is worth five and there
are 100 questions, the lowest score possible is 0 and the
highest is 500. For a student guessing randomly, the
expected score would be ___________ give or take
_____________ (the second number is the standard error of
estimate).
 d) What is the probability that this student who is
guessing randomly would get 150 or more points on this
test? (Show how you are getting your answer!)
4. A box has these 7 tickets: 0, 1, 2, 3, 4, 5, 6
 a) What is the average of the box?
________________
 b) Suppose we draw from this box 100 times, each time
replacing the ticket. If we got a sum of 275, this sum
would be composed of an expected value of ______________
plus a chance error of _____________.
 c) If we draw from this box 900 times with
replacement and total the tickets, the total should
behave as a normallydistributed variable with an
expected value (or average) of 2700 and a standard
deviation of 60. Given this information, what percentage
of the time would our total be between 2600 and 2800?
______________________________
 d) What is the probability that it will be between
2800 and 3000? _______________
 e) What is the probability that it will be greater
than 3000? _______________
 f) If we draw from this box 900 times with
replacement and total the tickets, the total should
behave as a normallydistributed variable with an
expected value (or average) of 2700 and a standard
deviation of 60. Given this information, 50% of the time
our chance error will have an absolute value less than
_______________________.
5. You have isolated one suit of cards (13 cards) from a
deck of cards to play a new game in which you play the role
of the house. If the player pulls out a J, Q, K or A, you
pay him $2.00. If the player pulls out any of the other 9
cards (2 through 10), he pays you $1.00.
 a) If we put this model into the terms of the box
model, what 13 tickets are in the box?
b) What is the average of the box?
c) What is the standard deviation of the box?
d) If people play this game 100 times with you, you
should expect to win $_______ give or take about
$___________.
e) If people play this game 10,000 times, you should
expect to win $_______ give or take about $_________
6. A box with a million tickets has an average of 60 with
a standard deviation of 5. If a sample of 400 draws is made
from this box, we expect an average of ______________ give
or take ____________or so. The probability that the average
would be less than 59 is ________________________.
7. If we draw 200 times from the box of tickets, 1, 1, 4,
we should get a total of _____ give or take about_____. This
second number is the standard error of the sum. We expect an
average of ______ give or take about ________. (This second
number is the standard error of the mean.)
8. A box contains these five tickets: 1, 3, 4, 5, 7
 a) Compute the average of the box. _____________
b) Compute the standard deviation of the box.
____________
c) If we draw, with replacement, 144 tickets from this
box, the expected value of the sum is ______________
d) If we draw, with replacement, 144 tickets from this
box, the standard error of the sum is _____________.
e) What is the probability that we will get a sum greater
than 600 when we draw 144 tickets from the box?
9. Four hundred draws will be made at random with
replacement from the box: 1, 3, 4, 5, 7.
 a) We expect the sum to be _____________ give or take
about _____.
 b) Estimate the chance that the sum will be greater
than 1550.
 c) We can expect the number 7 to be drawn ________
times give or take ____________.
 d) Estimate the chance that the number of 7s will be
greater than 85.
10. A box contains these five tickets: 0, 0, 0, 0, 5
 a) What is the average of the box?
b) What is the standard deviation of the box?
c) If a person draws 400 tickets from this box with
replacement, we would expect a total of _____________
d) The standard error of the sum when drawing 400 tickets
with replacement from this box is
____________________
e) What is the probability that the sum of draws will
exceed 370?
11. The box of three tickets, [ 1, 1, 4, ] has an
average of 2 and a standard deviation of the square root of
two. That means that if we draw two tickets out of the box
with replacement, we should have a new distribution with an
average of 4 and a standard deviation of 2. Show that this
conclusion is correct by computing that box of 9 tickets
that is equivalent to the two draws from the original box
and compute the average and standard deviation of this new
box.
Answers here.
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