Math 151 , Day 30, Friday, Nov.2, 2007 hit reload....After class.

HW Day30  PLEASE read Chapter 11 (pp. 286-291 optional). First pp. 271-77   Check p. 294: 11.17, 18 (parameter/statistic, sampling dist.).  11.19, 20 (behavior of xbars, mean & s.d.)  11.22, 23, 24 (behavior of xbars, more)  Read ahead:  Skip Ch. 12, 13.  Do Ch.14 on.
Memorize the 3 yellow-headed boxes on p. 278, 281 (mean and s.d. of sampling dist. of X-bar, Normality & Central Limit Th.)

Hand in Monday:
p. 275, 11.4 means in action (LLN)
p. 275, 11.5 insurance (LLN)

DISTRIBUTION OF XBAR(S) 
--These problems use only the mean and standard deviation.   
  p. 280, 11.7 (Teen cholesterol )
  p. 280, 11.8 (lab measurements)  For (b) they mean "what should n be?'

Postpone the rest, but PLEASE continue to read Ch 11, read over all the remaining HW problems (all are on Day 29) to get used to the words, questions, language here.  This is THE BIG IDEA chapter for the remainder of the course!
--These problems use  the "sample mean of n independent observations from a normal distribution has a normal distribution." theorem (p. 278) 
  p. 280, 11.9 NAEP math scores  (n = 1, n = 4)
  p. 290, 11.37 and 11.39 Pollutants in auto exhausts  For 11.39:  You might want to know L so that if you tested your 25 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. It is more reasonable to believe the exhaust system is not working, than that we "are" that 1 in 100 possibility."
  p. 289-90 11.36 and 11.38 Glucose testing  If we use this cutoff level L to say that people (with a mean of 4 tests) over L "have diabetes", then the chances of declaring that someone "has diabetes" when they really are OK (with mean 125mg/dl) is .05.  .05 or 5% is the chance of a "false positive" using this protocol, when the real mean is 125.
--These problems use the Central Limit theorem (p. 281) 
  p. 185, 11.10 What does the CLTh say?
  p. 286 , 11. 12 SAT scores, n = 1 and 70
  p. 286, 11.13, insurance (Hint: find P(Xbar> $275))
  p. 298, 11.41 auto accidents
  p. 298, 11.42 airplane overloads  (Hint: to do the problem you have to assume all the seats are taken.  Maybe not a reasonable assumption, but if there are empty seats, there's likely not a problem with overweight.)

Read, 
to discuss		 Optional 
 
Exams not finished.  Sorry again!  Monday for certain!
Hand in now, your sheet of results from 10.55 and 10.56 (Probability applet results). Add your 3 means from the samples of 4 grades (11.6)to the circulating list.
- HW questions? Day 29 
Continue with Ch. 11, Details Day 29
..Recap of Ch. 11: How do sample means behave?

                 Sample Chosen from a  Population
                  (varies)            (fixed, but usually unknown)
Calculate Numerical summary: Statistic estimating Parameter
                                    xbar                   µ
We take a simple random sample of size n, find the sample mean xbar.  It will be different depending on the sample, so we have a
                  random variable X bar.

Law of Large Numbers (p.273-4, "LLN")  Take observations at random from a population with population mean µ. Then as the number of observations n increases, the sample mean xbar gets closer and closer to µ. (Even if the population is infinite!   (Note--we don't say how big n needs to be for how close here.)

Now:  keep a fixed sample size n:
What is the distribution of the random variable Xbar, when the experiment is to take a simple random sample of size n? This is the distribution of means of all possible SRS's of size n.
We'll call it the "sampling distribution of the (sample) mean" (p. 275-7, then details 278-86)

We talked about the above; We'll do more examples using the normal shape of Xbar next time.

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.


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