Math151 Day29

Math 151 , Day 29, Wednesday, November 1, 2006 hit reload...After class

HW Day29 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. Do 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.)


*Read Ch. 11, to p. 286. Read it again. Try the "Check problems." Read the HW problems below and think about what you need to know to do them* (go ahead and do any that you can) *. Read the book Again. Formulate questions about what you don't understand! Bring to class.* 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 Friday: Sampling experiments which were due today (10.55 and 56, 11.6, as modified. see day 28) = = = = = = = = = = = = = = = = = = Postpone the rest; but READ all the problems, consider what you don't know how to do, as prep for Friday's class. 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 not quite finished. Friday for sure (I hope)
Add your 3 xbars from #11.6 p. 277 to the circulating yellow pad.
Hand in now, your sheet of results from 10.55 and 10.56 (Probability applet results)
- HW questions? Continuous Random Variables. Normal Random Variables. Day 28, Notes Day 28 Statistic/parameter
-From p. 277, 11.6 sampling distribution of exam scores Add your 3 means to the circulating list.
= = = = = = = = = = = = = = = = = = = = = = =
Class was used up on HW questions.   Continue from here Friday.
Continue, toward behavior of sample means: Details Day 28
Recap: How do sample means behave?
      Sample (varies) Chosen from a Population(fixed, but usually unknown)
Numerical summary:
               Statistic    xbar                    Parameter 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 phenomenon. We measure the outcome as a number, the sample mean, 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)

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)

The mean of the x-bars = the mean of the population

The standard deviation of the x-bars =

the s. d. of the population divided by the square root of n.

Consequences:

* X-bar "hits" the population mean on average--is "unbiased estimator" of µ (doesn't systematically go too high or too low.)

* Averages are less variable than individual observations. Averages from large samples are less variable than averages from smaller samples (because of dividing by the square root of n)

IF the population is Normal, the sampling distribution of Xbar is Normal.
"The Central Limit Theorem"
In any case, for "large" n, the sampling distribution of Xbar is Approximately Normal.

Example: "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 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) 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?

SPSS simulation: average of spinners which can land on any number between 0 and 1.
Population--one spinner. distribution flat between 0 and 1, mean .49 s.d. = .29
n = 2, Average of 2 spinners is Xbar. Distribution triangular between 0 and 1, mean .50, s.d. .21. .29/sqrt(2) =.205
n = 4, Average of 4 spinners is Xbar. Distribution normalish between 0 and 1, mean .50, s.d. .15. .29/sqrt(4) =.145
n = 15, Average of 15 spinners is Xbar. Distribution normal between 0 and 1, mean .50, s.d. .09. .29/sqrt(15) =.076

Xbars from SRS:
   Mean of Xbars is mean of population.
   Standard deviation of Xbars is s.d. of population divided by square root of n.
   As sample size increases, sampling distribution of Xbars gets more and more normal-shaped.
                (Central Limit Theorem)

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.
Pictures on overhead.   Author's website applet, Central Limit theorem


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