Math 151 , Day 37, Monday, Nov. 24, 2008 .After class..Hit reload .

Exam 4's not finished--sorry!.
Final exam: Wed. Dec. 17, 9-12 a.m.  If this is a problem for you, please email me soon.  
   Alternative--Monday afternoon, Tuesday morning/afternoon?? (Email your possibilities; I'll pick one!)
  Full exam schedule is at   http://www.wells.edu/pdfs/finals.pdf
     Registrar's page with link to this and other good stuff: http://www.wells.edu/academic/regist.htm

"Statistics means never having to say you're certain."
Confidence interval Estimation made our best guess at an unknown population mean.
Testing will investigate a claim made that the unknown mean is actually a particular value.
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Ch. 15: "Significance tests use an elaborate vocabulary, but the basic idea is simple: an outcome that would "rarely" happen if a claim were true--is good evidence that the claim is NOT true." (p.363 top)

Need machinery to analyze less "obvious" results--build in effect of standard deviation and sample size

Shoeboxes (white and yellow slips): Take a sample of size 4 from each,  record, return numbers.
   I claim the mean value  for both shoeboxes is µ = 20.  Am I telling you the truth?  I can't remember for sure.  I do know that the distribution in the box is normal,  standard deviation is 4.
I do remember that if  µ is not 20, then it is greater than 20. µ > 20.
Take a sample of size 4, find xbar.   Once for each shoebox! (should have found xbar already)
How far from 20 is it?  far enough that I believe the mean is not 20??

<>Measure your xbar's distance from 20  in standard deviations of Xbar's. (That is, find z for xbar, assuming µ = 20. Note s.d. for sampling dist of xbar is 2 (why?) ).   Example:  If I got an xbar = 24,  z = (24-20)/2 = 2 s.d.'s above mean. 
Is this a far-out value of z? Look in the normal table to see how much probability is in the tail to the right of it--gives a measure of far-out-ness independent of distribution ("P-value").   Prob. above 2 is about half of 5%, or .025, more exactly (1-.9772) = .0228.   IF the mean is really 20, I would see an xbar as high as 24 (or higher) about 2 to 3 in a hundred times.  So xbar = 24 is pretty strong evidence that real mean isn't 20.
Your shoebox results:  Do (one of) yours now, if you haven't. If you have, look around for someone who needs help.

Write your xbars (if you haven't) , z's, p-values, p<.10  (one on each paper--yellow or white) and make a dot for each on the circulating dotplot.

See Day 35 for the rest of the notes:  Brief overview:
The game:
Before taking data, define
H₀: "Null hypothesis" A claim or statement about the population we would like to show is NOT true.
   Stated usually as:  A parameter = a particular value.  H₀: µ =1000 hrs.  ("Average lightbulb life".)     H₀: µ =20 (shoebox mean=20)
H_a: "Alternative hypothesis" A claim or statement about the population we are trying to find evidence FOR.
    Stated usually as: The parameter  is >, or <, (one-tail tests) --  or NOT = the particular value. (two-tail)
    H_a:   µ  > 1000 hrs.   (Or   H_a:   µ  < 1000 hrs.  Or H_a:   µ  Not = 1000 hrs.)
    H_a: µ >20 (shoebox mean >20)

Take data.  Calculate test statistic, usually based on one that estimates the parameter in the hypotheses.  For µ, test statistic is the z-score of xbar, so a big z-score number means that xbar is far from µ.
    Is it an unlikely result if  H₀ is true?  Then that is evidence against H₀.

Measuring the strength of the evidence against H₀ (a common measuring stick for all distributions and parameters):
P-value of a test:  The probability, computed assuming that H₀ is true, that the observed outcome would take a value as extreme or more extreme than what we actually observed (if we could repeat taking-data again).  p. 368. 
Size of tail(s) farther out than the observed value.
    The smaller the P-value, the stronger the data's evidence against H₀ ( for H_a).

For a test of µ  , using xbar (sigma known), the P-value is
--the area of the tail beyond the observed xbar, in the direction of H_a (one tail)
(--or twice that area (two-tail).) We'll go over two-tail P-values in more detail after doing one-tail.
We usually calculate it by standardizing the observed xbar (assuming H₀ true) and looking in the normal table. (p. 369 on)

<>Applet:  P-value of a test of significance automates this.  (Uses "raw" scale of xbars, rather than z-scores).  How to. Use as check, guide. 
                 
Continue with new material ..!
Start with understanding "null and alternative hypothesis, p-value."   Those are the foundation. Then

A "Significance level" alpha is a probability level we decide on  in advance as being the "rarely" amount that will push us over into believing (well, sort of) that the H₀ claim  is not true. (Historically older language than P-value)
We tend to use simple benchmark numbers for it, like .10 (1 in 10), .05 (1 in 20), .01 (1 in 100).
When the P-value is less  than (or equal to) a particular significance level alpha (say .05), we say,
    "The results are significant at the alpha = .05 level," or "The results are significant (P< .05)" .  Giving actual P is better, if you can.
 Lightbulbs:  One-sided:  .0228 = P-value.  More than  2% and less than 3% chance of getting a result this far out (in this direction) if we did it again.
          "Significant at the alpha =.03 level.  Also at the alpha = .05 level"  (P-value says,  rarer than these levels)
          "Not significant at the alpha = .02 level.  Also not significant at the alpha = .01 level"  (P-value says, more common than these levels)
   Two-sided:  .0456 = P-value.  (Barely) less than 5% chance of getting a result this far out if we did it again.
            "Significant at the alpha = .05 level. (Also at alpha = .10).   Not significant at the alpha = .04 level.  Nor .01 level.
Applet:  Statistical Significance
You can pick the alpha you desire, and see if your x-bar lies outside the "alpha" barrier(s). (approach of p. 376-79) But P-value is more informative.
 A particular scientific discipline may have a commonly accepted set of benchmarks, and language to go with it. (I think I remember .05 = "significant", .01 = "highly significant" in psychology?)
We will be less doctrinaire, use the language "significant at the alpha = ___ level." 
(However, "nobody" uses a significance level less rare  than .10, 1 in 10).

We finished this; the Internet connection failed so couldn't demonstrate the Applet. Next time, HW repair, using the Applets, more on P-value/significance, and using Table C. "In practice" (Ch 16) things to watch out for.