Answers: Drawing Tickets from a Box Part
1
Back
to Problems
1. Suppose we have this box of tickets: 0, 2, 3, 4,
6.
- a) What is the average of the box?
3
b) What is the standard deviation of the box?
2
c) If we take a draw of 100 tickets out of the box
(replacing the ticket after each draw), we expect a sum
of ___300___
give or take
___20_____.
d) If we take a draw of 100 tickets out of the box
(replacing the ticket after each draw), we expect an
average of
_3____ give or
take ___ .2 or
1/5____.
e. If we draw 100 tickets out of the box and get a sum or
350, our chance error is ____
50 _____.
2. A box has four tickets, 1, 2, 3, and 10.
- a) What is the average of the box?
_____4
b) What is the standard deviation of the box?
___square root of 12.5 or
about 3.54
c) If 200 draws are made from the box, we expect to get a
total of about
___800___ give
or take about
____50__.
d) If 200 draws are made from the box, what is that
probability that the sum will be 875 or greater?
P(z>1.5) =
.668
3. A student is taking a multiple-choice 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 _______
1 ____
- b) The standard deviation of the box is
__________2
(big number minus the little
number times the square root of the product of the
fraction of little numbers times the fraction of big
numbers, or 5*(square
root((1/5)*(4/5))
- 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
____100____
give or take
_____20____
(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!)
-
- 150 is 2.5 standard
errors above the mean. According to the Normal
distribution, the probability of being that far above the
mean is .0062
4. A box has these 7 tickets: 0, 1, 2, 3, 4, 5, 6
- a) What is the average of the box? ____
3 ____
- 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 ____
300 ____ plus
a chance error of__
-25 ____.
- c) If we draw from this box 900 times with
replacement and total the tickets, the total should
behave as a normally-distributed 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?
.9044
- d) What is the probability that it will be between
2800 and 3000?
.0478
- e) What is the probability that it will be greater
than 3000? much less than
.001
- f) If we draw from this box 900 times with
replacement and total the tickets, the total should
behave as a normally-distributed 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
__________________. 50% of
the time the values will be within .6745 standard errors
of the expected value. Multiplying .6745 by 60 gives
40.47.
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 $_________
a) Four tickets with -$2.00
and nine tickets with $1.00. b) $1/13 or about $.077 c)
3*(square root of (4/13)*(9/13) or about $1.385. d) $7.77
give or take $13.85. e) $769.23 give or take
$138.46.
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 ____
60 ______ give or
take __ .2 _____or so. The probability that
the average would be less than 59 is
___less than .0001 because it is
five standard deviations below the expect
value.
7. If we draw 200 times from the box of tickets, 1, 1, 4,
we should get a total of _
400 _ give or
take about_ 20 _.
This second number is the standard error of the sum. We
expect an average of __
2 __ give or take
about __ .1 __.
(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. __
4 _____
b) Compute the standard deviation of the box. __
2 _____
c) If we draw, with replacement, 144 tickets from this
box, the expected value of the sum is ___
576 _____
d) If we draw, with replacement, 144 tickets from this
box, the standard error of the sum is ____
24 ____.
e) What is the probability that we will get a sum greater
than 600 when we draw 144 tickets from the box?
z = (600 - 576)/ 24 = 1;
P(z>1) = .1587
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
____1600____
give or take about
40 __.
- b) Estimate the chance that the sum will be greater
than 1550. P(sum >1550) =
P(z>-1.25) = .8944
- c) We can expect the number 7 to be drawn ___ 80 __
times give or take ___8_____.
(The box is a zero-one box
with one one and four zeros. The standard deviation of
the box is the square root of .2*.8, or .4. Multiplied by
the square root of n gives 20*.4 = 8.)
- d) Estimate the chance that the number of 7s will be
greater than 85.
- P(sum>85) =
P(z>(85-80)/8) = P (z>.625) =
.2660.
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?
-
a) 1; b) 2; c) 400; d) 40;
e) P(z>-.75) = ,7734
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.
The two tickets with a 1 are
separate, which gives these possible ticket draws, all with
equal probability: 11, 11, 14, 11, 11, 14, 41, 41, 44.
Adding them gives four tickets with two, four tickets with
five, and one ticket with eight. The total of the nine
tickets is 36, so the average of this new box is 4.
Subtracting the mean and squaring the differences gives a
total of 36. Dividing by nine and taking the square root
gives 2.
Back
to Problems
Go
to Part 2
|
|