Math 151 , Day 30, Friday, November 3, 2006 hit reload...After Class

HW Day30  Finish Chapter 11 (pp. 286-291 optional). First pp. 271-   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)  Next:  Skip Ch. 12, 13. Start reading 14.
Memorize the 3 yellow-headed boxes on p. 278, 281 (mean and s.d. of sampling dist. of X-bar, Normality & Central Limit Th.)
Quiz Monday:  see note below the HW box.
Hand in Monday:
p. 275, 11.4 means in action (LLN)
 p. 275, 11.5 insurance (LLN)

DIST. 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?'
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  finished.  Very good!  Exam 3 comments
Quiz Monday:  Like this:  The population has mean 125 and standard deviation 18.
You take a simple random sample of size 9.  The distribution of all possible sample means from such samples has
mean _____ and standard deviation______
Answers:  Mean is 125,
 standard deviation is 18 divided by the square root of 9.   Square root of 9 is 3, so standard deviation is 18/3 = 6.
that's all.
Hand in now, your sheet of results from 10.55 and 10.56 (Probability applet results)
-From p. 277, 11.6 sampling distribution of exam scores Add your 3 xbars from #11.6  to the circulating yellow pad. 
= = = = = = = = = = = = = = = = = = = = = = =
Behavior of sample means:  Details Day 28,  Day 29
Recap: How do sample means behave?
       Sample (varies)  Chosen from a  Population(fixed, but usually unknown)  
Numerical summary:
               Statistic    xbar                                Parameter  µ
                                                                                     
Sample mean xbar.  It will be different depending on the sample, so we have a random phenomenon.  
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. ( 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)

 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)

 So we can answer questions about the probability that Xbar will  fall in any particular region, if we know  n, and the mean and s.d. for the population (for X)

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