"It is the high privilege and sacred duty of those now living to educate their successors and fit them, by intelligence and virtue, for the inheritance which awaits them." -- (James A. Garfield at his inauguration, March, 1881) James Abram Garfield (1831-1881): Garfield was sworn in as the twentieth United States President on March 4, 1881. His time in office was brief since he was shot by a disappointed office seeker on July 2, 1981, and died from the wounds on September 19, 1981. Garfield is the only U. S. President to develop a proof of the Pythagorean Theorem. History records over 100 different proofs of this Theorem. Garfield's proof is quite eloquent, and easy to understand. What did Herkimer say to the monk who was always talking about different religious orders? Answer: "Can't you talk about anything except sects?" Things Herky would like to know: If the price of duck feathers increases, do you say that down is up? When nylon stockings were first sold in the thirties, was there a run on them? ASSIGNMENT #48 Reading: Section 8.1 (pages 429-430). Formulas on page 430 are extremely important... and they appear on the AP Statistics formula sheet. Exercises: 8.8, 8.9, 8.10 (page 427), 8.15, 8.16, 8.17 (page 431)...You can write responses in your text. Do this neatly. (You may do it on regular paper if you wish, but it would be nice to have the answers next to the statement of the problem.

You are in Section 8.1.

Suppose you roll a fair die 60 times. Let X = thenumber of 6's obtained. The possible values for X are0,1,2,3,...,58,59,60. Note that there are 61 (not 60) possible valuesof the random variable X.

This is a binomialsetting. There are 60 slots

- - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - -

and the probability of filling any slot is 1/6. Wehave independence, since knowing the result of one roll tells you nothingabout the result of the next roll.

In the case, N = 60 and p = 1/6. mx = Np = 60(1/6) = 10 andsx = sqrt[Np(1-p)] =sqrt[60(1/6)(5/6)] = 1.1785. From an intuitive sense, if you rolled afair die sixty times, you would "expect" to get ten 6's. Note that 10is the mean of the random variable X. Some other questions relatingto this binomial setting:

What is the probability that you wouldget exactly ten 6's?

Answer: 60C10(1/6)10(5/6)50 = binompdf(60,1/6,10) = 0.1370 =13.7%.

What is the probability you would get five orfewer 6's?

What is the probability that you would not get a 6in the sixty rolls?

Answer: 60C0(1/6)0(5/6)60 = (5/6)60 = binompdf(60,1/6,0) =0.000017747 = 0.0017747%

Note that 0.000017747 is approximately 1/56348. Inother words, if we define an event to be rolling a fair die sixtytimes, then you would expect to get no 6's once every 56,348events.

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Text:
The Practice of Statistics, by Yates, Moore, McCabe. New York,W.H. Freeman and Company, 1999. (ISBN 0-7167-3370-6)

Supplemental books:
The Cartoon Guide to Statistics, by Gonick and Smith. NewYork, HarperCollins Publishers, 1993. (ISBN 0-06-273102-5)
How to Lie with Statistics, by Darrell Huff. New York, W.W.Norton & Company, 1982 (ISBN 0-393-09426-X)