"The most distinct and beautiful statements of any truth must take at last the mathematical form." -- (Henry David Thoreau, 1817-1862) Numbers have interesting historical backgrounds, and there are many interesting stories behind just about any counting number. Here are just a few very brief tidbits relating to numbers. Some research will yield many interesting stories. Previous Tidbits have referenced the digits 1,2,3,4,5,6,7,8,9. Here are just brief statements about some other numbers. 10: The number 10 = 1+2+3+4. Since the ancient Greeks saw "four-ness" in many things, they considered 10 the ideal number. 17: Not notably important in Christian tradition, but quite important in the ancient Near East. The Muslim alchemist Jabir ibn Hayyan saw the entire material world based on 17, since he claimed the series of numbers 1,3,5,8 (which sum to 17) formed the foundation of all other numbers 20: This number formed the basis for counting in some ancient cultures. After all, if we count our fingers and toes, we get 20. The Maya used base 20. 33: A number of completion and perfection. In Christian tradition, Jesus lived on earth for 33 years, and David ruled for 33 years. 40: Of the double digit numbers, 40 is one of the most fascinating. It appears quite frequently in history. Lent lasts for 40 days, the Biblical flood lasted 40 days, the 40 large stone pillars in Stonehenge are arranged in a sacred circle with a diameter of 40 steps, the children of Israel wandered in the desert for 40 years, etc. What did Herkimer learn when he tried to light a fire in his kayak? Answer: You can't have your kayak and heat it too. Herky's friends: JACK O'LANTERN...this guy worked in a pumpkin patch. AL E. BYE... he had an excuse for everything. ASSIGNMENT #61 Reading: SUMMARY(pages 527-528). Section 10.2 (pages 531-539). There is a good "Outline of a test" summary on page 536. Exercises: 10.27, 10.28 (page 536).

You are working with ideas and concepts fromSections 10.1 and 10.2.

Example:

A production line population P is known have mean= .28 and standard deviation = .03. (m = .28 and s = .03 are populationparameter.) A random sample of size 50 is chosen and the sample meanx(bar) = .272.

Question: Based onthe sample mean, is it reasonable to conclude that the sample camefrom P?

Null hypothesis H0: m = .28.

Alternate hypothesis Ha: m < .28. (If true,production line standards aren't being met.)

The statistic being tested is x(bar) = .28. IfH0 is true,then the Central Limit Theorem tells us that x(bar) is a number in anormal distribution with mean = .28 and standard deviation =.03/sqrt(50) = .0042. The probability that a sample mean would beless than .272 is normalcdf(-1E99,.272,.28,.0042) = .028405, or about2.8%. In this case, our P-value is 2.8%. This is theprobability that we would get a sample mean as "extreme" as .272 ifH0 is true.

Did our sample come from P? That is, can weconclude that the stated production standards are being met based onour analysis of the sample mean? Realize, of course, that we can'tgive a definite answer to these questions? If we rejectH0 andconclude something is wrong, there is about a 3% chance we will makea TYPE I ERROR. (That is, there is about a 3% chance that we willincorrectly include that H0is false when it is actuallytrue.)

For now, let's just attempt to understand how toset up a statistical test, and how to calculate a P-value.

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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)