Math 151 , Day 32, Wed. Nov. 12, 2008 hit reload....After class .

•  Whatever the population distribution of X, that we draw the sample from, (see p. 278) 
     (as long as the population is large compared to the sample (at least 10 to 20 times sample)
  The mean of the x-bars = the mean of the population
  The standard deviation of the x-bars =
     the s. d. of the population divided by the square root of n.
• IF the population is Normal, the sampling distribution of Xbar is Normal.
• "The Central Limit Theorem"
  In any case, for "large" n, the sampling distribution of Xbar is Approximately Normal.
  

HW questions

Closed book Quiz  now. ( Work on the paper handed out, fold once, hand forward.)  Solutions
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Central Limit Theorem...
How large is "large"?  How approximate is "approximate"?
    If the population was close to normal, n doesn't need to be very large.
    If the population is not badly skewed or bimodal, n=25 already gives a pretty good approximation to normal.
SPSS simulation: average of  spinners which can land on any number between 0 and 1. (Shown Mon.)
Author's website applet, Central Limit theorem for a highly skewed dist.
Sum of 3 dice (divide bottom scale by 3 to get average of 3 dice)
Pictures, n=1, 2, 4, 25   CLTpics.pdf  
Rice U. Applet 

What if the population is not 10 to 20 times the sample size?  The real s.d. of the x-bars will be narrower than the rule above.  You may not "get to" normal as a shape.  Sample of 4 grades from a population of 10:This and one other semester (big file, loads slowly off campus)  last year  All possible samples.

Do this Friday
"Fuzzy Central Limit Theorem:"
Data whose variation is due to adding  many   small    independent   random influences will have an approximately normal distribution.
  Balls and pins, heights of women, etc.  (p. 281, after the yellow box)
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Chapter 14, beginning:
 SAMPLE from an UNKNOWN population.  Each person took 4 slips from the Birkenstock box,  for
  HW:   found 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???

 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.   It's a "Confidence interval" estimate.
      You Recorded them on the sheet going around, and drew the interval on the graph  going around.
         If xbar = 8.0       7.159|_____________8.0_____________|8.841
Remember: xbar is the statistic that estimates the parameter µ

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. Most important, now and forever.  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)   (Each mean 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."

Previous years: 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).


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)

• the estimate  is xbar
• margin of error m is :  z* times Standard deviation of sample mean
z* from Normal table.  Probability C is between -z* and +z*.

Standard deviation of sample mean:  Sigma /sqrt(n)
Must know standard deviation of population!

  m = z* (sigma)/ sqrt(n), so CI is xbar +  z* (sigma)/ sqrt(n)
.