Math 151 , Day 32, Wed. Nov.7, 2007 hit reload....After class.

HW Day30  (Re)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)  Next:  Skip Ch. 12, 13.  Start reading Ch.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.)

Hand in , finally

DIST. OF XBAR(S), cont'd.
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.
Note:  11.38 and 11.39 are "backward" Normal distribution problems:  going from proportion/probability to x (here L).

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 returned last time: get yours from me.

You took 4 Numbers (random sample) from the Birkenstock box:  Found mean xbar.  Found xbar + .841.  This is your interval estimate of the unknown mean of the box's population.  ("margin of error" is .841) (Returned your numbers afterward.)   (Chapter 14)
Add your values to the list, and graph your interval on the graph circulating.   Call out if you fill up the graph: I have another.
     If xbar = 8.0       7.159|_____________8.0_____________|8.841
Quiz after recap.

<>~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Re-RECAP: 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 worked on computations using the sampling distribution of the mean Monday: Here's what we were doing (fixed and finished)
"Normal" body temperature 98.6 deg. on average.  (Assume this is true.)
Assume normal distribution, & s.d.among many people is 0.6.
   Probability that one (random) healthy 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?

All of these are P(Xbar>98.8) for different sample sizes n.  (Normal table A)

Sample size n
 s.d. of Xbars =  (pop.s.d.)/sqrt(n)
 z = (raw-mean)/s.d.
 P(Xbar>98.8)=  P(Z>z)
1
 .6/1 = .6
 (98.8-98.6)/.6 = .2/.6 = .33
 P(Z>.33) = .3707
4
 .6/2 = .3
 (98.8-98.6)/.3 = .2/.3 = .67 P(Z>.67) = .2514
36
 .6/6 = .1
 (98.8-98.6)/.1 = .2/.1 = 2 P(Z>2) = .0228
100
 .6/10 = .06
 (98.8-98.6)/.06 = .2/.06 = 3.33 P(Z>3.33) = .0004

Closed book Quiz  now.
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Returning to HW problems, from above list.  Summary, demonstration links

Do this Friday:
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.

What if the population is not 10 to 20 times the sample size?  The real s.d. of the x-bars will be narrower than the rule above.  You may not "get to" normal as a shape.  Sample of 4 grades from a population of 10:  This year  All possible samples.
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