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Answers: Drawing Tickets from a Box Part 1

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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.

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