Math 151 , Day 27, Friday, April 6, 2001

Fuzzy Central Limit Theorem:
Data whose variation is due to many small independent random influences will have an approximately normal distribution.

Questions on HW: Probabilities involving sample means.
SAMPLE from an UNKNOWN population.  Each person take 4 slips from the Birkenstock box,
    find the mean, and the mean + .841.
   Record these for yourself & on the sheet going around, and draw the interval on the graph transparency going around.

Inference: beginning notes, Day26
Confidence interval estimate of a(n unknown) population parameter:

 Confidence Interval of the form  estimate + margin-of-error  for the mean, Confidence level C: (p.306) 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?

Why does the formula work?

  1. If a particular xbar is within m of the population mean, then the interval xbar + m contains the population mean.

    2. & If a particular xbar is farther than m from the population mean, then the interval xbar + m doesn't contain the population mean.
  2. We choose z* (and from it m) so that the probability that Xbar is within m of the population mean    is C.

    3. How? Probability C is between -z* and +z* in the standard normal table,
        between -z* (s.d. of Xbar) and +z*(s.d. of Xbar) in the normal distribution of Xbar.
  3. Table C, bottom row, is a restating of  table A, normal table, with probabilities (areas) on the edge, and z values in the body.  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.  Table C gives it more precisely as  .841.
Some assumptions:
SRS
Xbar is normal!  OK IF 1) population is normal, or 2) n big enough for Central Limit theorem.
Sigma for population is known.  Rarely true in practice.  Large n?  substitute s calculated from sample. Small n--Ch. 7.

HW Day27,   Read (and reread) Sec. 6.1  I'll try to finish it Monday.
Memorize the definition of a C.I. p. 302, esp. the "repeated samples" bit, and the formula p.306 for a CI for the mean..
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.
Hand in Monday, HW Sec. 4.3 (assigned Day 26)
Prepare the following, and bring questions on Monday.  All will be due (and possibly more) on Wednesday
Prepare for Monday, to Hand in Wednesday: Sec.6.1  . 
p.302, 6.1 poll of women
6.2 95% confidence?
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. 

Using formula p. 306 for C.I.: 
6.6 potassium again.
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 Read pp. 312-13 before doing this one. 

Cautions; general review and extension 
p. 314 6.14 internet, response rate
p.317, 6.19 newts
p. 318, 6.22  men/women CI's
p.316, 6.18  consumers/pharmacies		 Read, to discuss 
 
 
 
 
 
 
 
 
 


 
 
 
 
 


6.13, 6.15 (cautions) 
6.23 (Carter election)		 Optional
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