Math 151 , Day 30, Friday, April 16, 2004 hit reload..

HW Day30  Sec. 6.1 Read it again.
Memorize the definition of a C.I. p. 302, esp. the "repeated samples" bit (and where z* comes from), or below,
                 and the formula p.306 for a CI for the mean (or below).
    Closed Book Quiz Wednesday   on the two Confidence Interval things.
Note on reading:  p. 306, table at top:  "Tail area" is area in One tail, which is what you look up in the Normal table.
Sec.6.1 : 
To be handed in Monday: 
p.302, 6.1 poll of women
6.2 95% confidence?
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Using formula p. 306 for C.I.: 
A.  Find the critical value z* for C = 84% (like example 6.3, pp.304-5)
6.6 potassium again. (a) n=1 (b) n = 3.
6.7 comparing CI's for different confidence levels.  Also write down the m (margin of error) for each interval. 
6.9 comparing CI's for different sample sizes.
6.5 IQ test scores. For b), Find xbar with a calculator.   Read pp. 312-13 before doing part c. 
= = = = = = = = = = = = = = = = = = = = 
Postpone:  6.3 density of x-bar, and confidence intervals This problem combines the pictures 6.2 and 6.4 For part d, to draw the confidence interval:  just choose any point on the horizontal axis of  your graph to be x-bar.  Measure off the distance m (half the width of the shaded interval) and extend a bar m wide to the left and the right of your point,below the curve.  (Like fig. 6.4, the bars with arrows at the ends.  The red dots show what the x-bar is for that confidence interval)  Choose another point, and repeat..  If your first x-bar was in the shaded interval, pick your second outside the shaded interval, and vice versa.  You should note that if x-bar is in the shaded interval, then the confidence interval bar covers mu (280) and if x-bar isn't, then the bar doesn't. 		 Read,
to discuss		 Optional
Chapter 6, If you didn't Wednesday:
 SAMPLE from an UNKNOWN population.  Each person take 4 slips from the Birkenstock box,
      find the mean, and your mean + .841.
      Record these for yourself .  This is your  "Interval Estimate" of the mean of the shoebox population.
          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.
      Record them also on the sheet going around, and draw the interval on the graph transparency going around.
         If xbar = 8.1       7.259|_____________8.1_____________|8.941
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Central Limit Theorem again.  HW questions?
How big does n have to be for Xbar to have a normal distribution? (about 25 is always good.)
"Fuzzy Central Limit Theorem:" Day 29
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Chapter 6, Introduction to Inference
Statistical Inference: drawing conclusions about a population from sample data.
    Requires: Random sample or Randomized experiment.  (Simple Random Sample usually)

First example:  Use sample mean xbar  to "estimate" (unknown) population mean µ
 Mean of 4 grades (HW#4.40) estimates population mean of all 10 ("known"= 69.4)  E.g. 69.75,  64.25,  73.5
(Each is a "point estimate")

Interval estimate:  xbar + margin of error (fudge factor)  estimates population mean µ (69.4)

    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:

Confidence Interval of the form  estimate + margin-of-error  for the mean µ with Confidence level C: (p.306) (Table A, or Table C, t dist. bottom row) z*?  Probability C is between -z* and +z* in the standard normal table.
To get the z* for C = 60% from the normal table, note that this is the middle 60%, which leaves 40% to be split between the 2 tails.  So 20% above z*,  and 80% below.  Go into the body of table A, find .8000 is between values .7995 and .8023, closer to .7995.  The z value with .7995 below it is .84.  z* = .84.   (Cf. example 6.3, pp.304-5)
Table C, bottom row, is a restating of  table A, normal table, but with probabilities (areas) on the edge, and z values in the body.   Bottom row gives the most common C's.  Read off, for C =60%,  z* = .841.

Example:  Sample of size 9 from a Normal population with unknown mean and pop. s.d. sigma = 6,  xbar = 12.
  Find a 90% CI estimate for the unknown mean µ:
n=4.     (sigma)/ sqrt(n) = 6/3=2
z* = 1.645,  so m = 3.290;
                       CI is 12 + 3.290, or  8.710 to 15.290.

The Birkenstock box contains numbers from a normally distributed population, with population standard deviation 2.
You each constructed a 60% confidence interval for the unknown mean:
    n = 4.
    Standard deviation of sample mean = 2/sqrt(4) = 2/2 = 1
    z* for C = 60% is .841, so margin of error m is .841 times 1= .841.
How many people captured the true mean?
( previous classes,11/20 = 55% ,  22/29= 76%.   9/18 = 50% , 11/20 = 55%,  15/22= 68%,  14/22 = 64% 
Combined, 82/131 = 62.6%    This class?
Quite variable for small numbers of samples, but settling down.)


Why does the formula work?

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