Wikipedia:Reference desk/Archives/Mathematics/2010 June 11

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June 11

Simple Type 1 error problem

I've got a statistics homework problem that I'm not sure how to approach. Here it is:

1) In most industrial procedures quality control is routinely performed. Imagine the following situation: a machine is used to fill two liter soda bottles. Even when the machine is calibrated to put two liters of soda in each bottle, the actual content varies a little from bottle to bottle. The content has a normal distribution with mean 2 liters and standard deviation 0.1 liter. Each day a random sample of 9 bottles is taken from the daily production. If the sample mean is below 1.9 liters or above 2.1 liters, the production process is stopped to check the machine.

a) What is the probability that the sample mean is either below 1.9 liters or above 2.1 liters?

b) What is the probability that the machine is stopped even though it is within working according to specifications? Explain in complete sentences.

Part a is simple; I got .0027 for being +/- 3 standard deviations from mean. I'm not even sure what part b is asking, though. If it's asking the probability that a machine has a mean within [1.9, 2.1], then isn't it the same as part a? Anyone know what's going on here? --151.141.244.19 (talk) 01:53, 11 June 2010 (UTC)

I won't say if your answer to part 'a' is correct (why is ±3 standard deviations the same as a sample mean of ±1 standard deviation from the desired mean?), but it looks to me like 'b' has the same answer as part 'a'. If the sample mean exceeds the required bounds, the machine is stopped and checked. ~Amatulić (talk) 04:42, 11 June 2010 (UTC)

Because a sample consists of 9 bottles, and the square root of 9 is 3. Bo Jacoby (talk) 19:59, 12 June 2010 (UTC).

I think part (b) is just asking you to put your answer from part (a) into a complete sentence, e.g. one beginning, "The probability that the machine is stopped even though ..." --Qwfp (talk) 19:39, 11 June 2010 (UTC)

Proof that Euclid’s fifth postulate cannot be proven.

How did Bolyai and Lobachevski prove that Euclid’s fifth postulate cannot be proven?91.104.181.154 (talk) 22:24, 11 June 2010 (UTC)

The fifth postulate was shown independent of the others by the demonstration of a model of hyperbolic geometry which satisfied the other postulates but not that one. Dmcq (talk) 23:11, 11 June 2010 (UTC)

By the way I recently went along to an exhibition of crocheted models of the hyperbolic plane. Here's the person who started it all off Daina Taimina. Dmcq (talk) 23:14, 11 June 2010 (UTC)

2.8651496649...

I've found this to be the "other" positive value of x for which $\Gamma (x)={\sqrt {\pi }}$. Does anybody know anything else about this number? --Lucas Brown 23:22, 11 June 2010 (UTC)

Well I tried out Plouffe's inverter and it didn't have it. Dmcq (talk) 23:56, 11 June 2010 (UTC)

You can do more tests using integer relation algorithms if you compute more significant digits (at least a few dozen). Count Iblis (talk) 03:13, 12 June 2010 (UTC)

Here are a few more digits:

2.8651496649764734274885554227037096412511096062528695651871023239515553871026286151412171720144439329280321651530519737995653560...

-- Meni Rosenfeld (talk)