Math 151 , Day 31, Monday, November 6, 2006 After class hit reload...

Jenn O'Neill writes:  Hours today from 3-4:45pm in the math clinic. My hours for the rest of the 
week will be the usual Tues 3:30-5:30 and Wednesday 6:30-8pm. I am also available for individual 
appointments. 

"Fuzzy Central Limit Theorem:"
Data whose variation is due to  many   small    independent   random influences will have an approximately normal distribution.
  Balls and pins, heights of women, etc.  (p. 281, after the yellow box)
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Chapter 14, beginning:
 SAMPLE from an UNKNOWN population.  Each person take 4 slips from the Birkenstock box,  write them down, return slips.
  HW:   find the mean, and your mean + .841.
     Your mean is your best guess at the real mean, based on your sample.  It's not going to be exactly right.  So you build in a fudge factor.
     Your  mean + .841. is your  "Interval Estimate" of the mean of the Birkenstock population.  Does it capture the real mean???

Next time:  Your "estimate" of the (unknown) population mean µ of the numbers in the shoebox is your sample mean plus or minus the "fudge factor/margin of error" .841.
      You'll Record them next time on the sheet going around, and draw the interval on the graph transparency going around.
         If xbar = 8.0       7.159|_____________8.0_____________|8.841

Introduction to Inference: Chapter 14, Confidence intervals
Statistical Inference: drawing conclusions about a population from sample data.
    Requires: Random sample or Randomized experiment.  (Simple Random Sample usually)

"Simple conditions" to develop concepts.
     --SRS.  No "difficulties", no bias   (Population is at least 10 to 20 times as big as sample)
    --Variable X is perfectly Normal, mean  µ, s.d. sigma.  (We'll extend from this later)
   --  µ is unknown, but sigma is known!  (we'll remove the sigma-known condition later)

First example:  Use sample mean xbar  to "estimate" (unknown) population mean µ
 Mean of 4 grades (HW#11.6) estimates population mean of all 10 ("known"µ = 69.4)  E.g. 69.75,  64.25,  73.5
(Each is a "point estimate")

• "xbar IS µ"   Never true exactly
• "xbar is close to µ"  True for most xbars, depending on "close", and sample size n.
• "xbar is probably close to µ" ["probably"?? It is or it isn't, with our definition of probability. We have "confidence" ours is a close one]
• "xbars are usually close to µ" True.
• If we have only one sample -- one xbar, we have NO IDEA if ours is one of the "close" ones.
• Quantify "usually" and "close."

Fall 2002: 33% (16 of 48) xbars  recorded were within 1 of µ. (between 68.4 and 70.4).
         83% (40 of 48) xbars  recorded were within 4 of µ. (between 65.4 and 73.4).
         94% (45 of 48) xbars  recorded were within 5 of µ. (between 64.4 and 74.4).

Fall 2001:  65% (15 of 23) xbars  recorded were within 4 of µ. (between 65.4 and 73.4).
                  78% (18 of 23) xbars  recorded were within 5 of µ. (between 64.4 and 74.4).
Spring 2002:  83% (35 of 42) xbars  recorded were within 4 of µ. (between 65.4 and 73.4).
                       90% (38 of 42) xbars  recorded were within 5 of µ. (between 64.4 and 74.4).
                                       (I excluded impossible xbars)
Note tradeoff between "close"(accuracy) and "usually" (confidence)

    69.75 + 1:   "µ is between 68.75 and 70.75"  True
    69.75 + 4:   "µ is between 65.75 and 73.75"  True
      73.5 + 4:    "µ is between 69.5 and 77.5"  False
      73.5 + 5:    "µ is between 68.5 and 78.5"  True
       64.25 + 4:   "µ is between 60.25 and 68.25"  False
       64.25 + 5:   "µ is between 59.25 and 69.25"  False

Confidence interval estimate of a(n unknown) population parameter: (pp. 346-7)

• an Interval constructed from the data,  (usually estimate + margin of error) +
• a Confidence level C: where
  C = probability that intervals constructed by this method will capture the true, unknown, parameter.
    (C is "success rate" for the method -- if you use the method repeatedly)