Math 151 , Day 26, Wednesday, April 4, 2001

Notes on Day 25
"The Central Limit Theorem"
In any case, for "large" n, the sampling distribution of Xbar is Approximately Normal.
(With, of course, mean of Xbar = mean mu of population,
s.d. of Xbar = s.d. sigma of pop. divided by sqrt of n.)
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.
    Even if the population is pretty weird, n=25 or 30 gives a pretty good approximation to normal.
Pictures on overhead.
SPSS simulation: average of  0-1 spinners.

Example:  "Normal" body temperature 98.6 deg. on average. Assume normal distribution, & s.d.among many people is .6.
  Probability that one  individual's  normal temperature is below 98.0 degrees?
  Take SRS of 9 people.  Sampling distribution of the mean?  Probability that the mean is below 98.0?
   Probability that one (random) individual's normal temperature is above 98.8?
   Probability that the mean of a sample of 4 is above 98.8?
   Probability that the mean of a sample of 36 is above 98.8?
   Probability that the mean of a sample of 100 is above 98.8?

Starting here, Day27
Chapter 6, Introduction to Inference
Statistical Inference: drawing conclusions about a population from sample data.
    Requires: Random sample or Randomized experiment

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

Interval estimate:  xbar + margin of error (fudge factor)  estimates population mean mu (69.4)
    69.75 + 4:   "mu is between 65.75 and 73.75"  True
    73.5 + 4:   "mu is between 69.5 and 77.5"  False
    73.5 + 5:   "mu is between 68.5 and 78.5"  True
    64.25 + 4:   "mu is between 60.25 and 68.25"  False
    64.25 + 5:   "mu is between 59.25 and 69.25"  False

Confidence interval estimate of a(n unknown) population parameter:

HW Day26, Wednesday, April 4  Reread sec. 4.3. (Skip ch. 5) Read Sec. 6.1 (again and again...)
Memorize the tan box on p. 242 (mean and s.d. of sampling dist. of x-bar)
Hand in FRIDAY
LLN: p. 234, 4.39 betting on the numbers
These problems use only the mean and standard deviation. 
  p. 243, 4.41 (lab measurements)
  p. 250, 4.50
These problems use either the Central Limit theorem, or the "sample mean of n independent observations from a normal distribution has a normal distribution." theorem (both on p. 244)
  p. 249, 4.51 cola (you did a, now do b) 
  p. 247, 4.44 carpet flaws.  Also draw some square yards and mark some flaws.
  p. 250, 4.53 auto accidents
more... Try now, Hand these in MONDAY
p. 243 4.42 unbiasedness, sample size
p. 249 4.52 hypokalemia
p. 249 4.48 dust Note, the dust actually weighs 123mg, but the weighings may not be accurate enough for us to find the actual weight. "Distribution of this mean" = "Distribution of means from all possible sets of 3 weighings from these scales." When I took physics, we did not have digital scales; they were balance beams; and we weighed everything 3 times and found the average. (Have you ever gotten on the scale, said "that can't be right!" gotten off and on again a couple times?) 
. 
p.250 4.54 (labeled 4.53) pollutants; backward from value to probability.  You might want to know L so that if you tested your 125 cars and found a high value of x-bar, you would be able to compare it with L; if it was greater than L, you would go back to the manufacturer and say "I  believe you sold me a batch of bad cars, because the chances of getting an average emission level this high if the exhaust system is working properly is only 1 in 100, and it is more reasonable to believe the exhaust system is not working, than that we hit that 1 in 100 possibility." 		 Read, to discuss Optional
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