Math 151 , Day 29, Wednesday, Oct.31, 2007 hit reload....after class.

HW Day29  Finish ch. 10: pp. 261-2.  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)  Next:  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 Friday:
If you didn't, Sampling experiments which were due today (10.55 and 56, 11.6, as modified.  see day 28)
Also for sure:  (you can do this one without much understanding of chapter 11:)
p. 277, 11.6 sampling distribution of exam scores Do a and a modified version of b; Do b this way.  Close your eyes and put your finger down somewhere on table B (Don't use row 116!! unless you land there.).  Start reading the table where your fingertip lands.  Record your sampleof 4,  and find xbar for your sample.
Now Repeat part b, to get a total of  3 values of xbar. (You can just keep reading the table where you left off, or you can put your finger in a different spot).  Record your 3 xbars,  Make a dotplot of your 3 xbars and bring the values to class to be compiled with everyone else's..

= = = = = = = = = = = = = = = = = = 
READ Personal Probability, pp. 261-2.  All our theory will be developed using the "frequentist" point of view (probability = proportion in the long run).  But there is another theory based on Personal Probability, sometimes called "Bayesian".
p. 262, 10.18

 PLEASE read Ch 11, read over all the HW problems to get used to the words, questions, language here.  This is THE BIG IDEA chapter for the remainder of the course!

= = = = = Ch. 11--= = = =
p. 272 11.1 caffeine (Param./Stat.)
p. 272 11.2 voters(Param./Stat.)

Postpone the rest:
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?'
--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 
    More practice on Normal:
p. 261, 10.17, ACT scores

- - - -
p. 272,11.3 Bearings (Param./Stat.)
Exams not finished.  Sorry again!
Hand in now, your sheet of results from 10.55 and 10.56 (Probability applet results)
- HW questions? Discrete, Continuous Random Variables.  Normal Random Variables.  Notes Day 28 
..
<Notes cover "all" of Ch. 11>~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
We know that a sample from a population will not exactly represent the population.  If we take a random sample, the behavior of samples will not be individually predictable, but there will be predictable pattern in many random samples from the same population.  Knowing the pattern will be  as good as we can do.
Ch. 11:
        Sample Chosen from a  Population
          (varies)             (fixed, but usually unknown)
Calculate
Numerical summary: Statistic (Latin) Parameter(Greek letter)
    Examples:           Sample mean xbar    Population mean mu (µ)
                       Sample st. dev. s    Pop. standard dev. sigma
                        Sample median     Pop. median
                Sample proportion p-hat  Pop. proportion p
                Sample line height y-hat  Pop. regression line height y
The actual value of the Statistic will vary, depending on the particular sample. "Sampling variability"
..
The Statistic "estimates" the Parameter.  We hope it is close to the parameter.  If we choose simple random samples, we can understand the pattern of values the statistic can take.
Some examples of  statistics:
    Height:   U.S. young women: pop. mean= 64.5", pop. s.d. 2.5"  (ed. 2 p.66.  BPS4e p. 76, 87 says women 20-29 is N(64, 2.7))
                                               Math 151, Spring '01,  xbar = 64.2,     s = 3.75.
                                                                        Fall '01,   xbar = 65.01,    s = 3.22.
                                                                     Spring '02,  xbar = 64.53,    s = 2.91.
                                                                       Fall '02,    xbar = 63.89,     s = 2.48.
                                                                   Spring '03,  xbar = 64.98,    s = 3.29
                                                                     Spring '04,  xbar = 65.33,    s = 2.25
                                                                       Fall '04,  xbar = 64.68,     s = 3.54
                                                                    Spring '05,  xbar =64.31 ,    s =2.93
                                                                         Fall '05  xbar =63.92 ,    s =2.80
                                                                    Spring '06  xbar =62.93 ,    s =2.78
                                                                        Fall '06  xbar =62.81 ,   s =  2.65
                                                                    Spring '07  xbar =65.18 ,    s =2.26
                                                                        Fall '07  xbar =65.67 ,   s =  2.73 
<>   Coin flip: Proportion of heads  p = 1/2 (?)       p-hat =  256/520 = .492  (combined data from many past classes)
    Thumbtack:  Proportion of point-up p =  (??)       p-hat =  441/691 = .6382  (one past class, Math 251)

Next:(Start here Friday) How does sample mean behave? ( pp.275-86)
                 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 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 infinitely large! Note--we don't say how big n needs to be for how close here.)
  OR Let the sample size n get bigger.  Then  the xbars will eventually get very close to the population mean µ.
  OR As the sample size increases, the sample mean gets closer to the population mean µ.
  OR For a very large sample, the sample mean will (almost certainly) be very close to the population mean.
e.g. the bigger my statistics class, the closer their mean height should be to the U.S. mean height for women.
(Statistics means never having to say you're certain...)

Applet: http://www.whfreeman.com/bps4e  "Law of Large Numbers"  Roll a single die.  X = number.  µ= 3.5. (Think X is number of spaces you can move in a board game.  Average per roll is 3.5.)
My result at home --1st roll: x = 5    n=1,  Xbar = 5/1 = 5. 
                          Roll again, x = 2.  n=2,  Xbar = (5+2)/2 = 3.5
                            Again,     x = 1   n=3,  Xbar = (5+2+1)/3 = 2.67  Again... ...   Xbar for large n; close to 3.5.

Now:  keep a fixed sample size n:
What probability distribution describes the random phenomenon of finding xbar from a SRS?
That is, what is the distribution of the random variable Xbar, when the experiment is to take a simple random sample of size n?
We'll call it the "sampling distribution of the (sample) mean" (p. 275-7, then details 278-86)
This is the distribution of means of all possible SRS's of size n. 
10.3b  "penny" p = .1  Your results, Previous classes, some of the possible sample proportions.  Simulation of sampling distribution of proportion.
If you have this R.V.:  X = 1 if Heads, 0 if Tails, then P(X=1) = .1, P(X=0) = .9.  And Xbar = (sum of all observations)/n (# of Heads)/n = sample proportion!  So it's also sampling distribution of mean!  Theoretical distribution
What do we see?
--Shape: Looks normal-ish
--Center:  Mean of xbars ~ mean of dist. of X.  (.1)
--Spread:  SD of xbars  is smaller than that of population X.  Applet: http://www.whfreeman.com/bps4e  Normal approx to binomial, p = .1, n = 1, then n = 100 (our data is for n = 200, even narrower.)  Horizontal scale is in n, number of possible heads, and we worked in p, proportion of heads.  Just think of  space between 0 and the given number as being from  0 to 1, black line (mean) is at .1, always.

..
HW tonite #11.6 (modified): each get 3 SRS's  of size 4, find 3 means: will pool to get histogram of Sampling distribution of mean
Quincunx board:  Result for one ball is "average" of going + or going - at each level
      (entry + pin 1+pin2+ ...+ pin 6).Continue, toward behavior of sample means:

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)

 Example: "Normal" body temperature 98.6 deg. on average.  (Assume this is true.) (Normal table A)
Assume normal distribution, & s.d.among many people is 0.6. (0.7 is a better assumption, I'm told, but .6 is easier to think with.)

We talked about the above Friday; We'll do the following examples using the normal shape of Xbar Monday.
  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.
  Author's website applet, Central Limit theorem for a highly skewed dist.
Pictures on overhead. 
Rice U. Applet  (kind of creaky) where you can change/create the population dist.  Sometimes it fails in pieces (sd=0) or crashes if you try to use all the options, but is pretty good if you stick to the Mean, and use only the top 3 displays, and don't go for huge numbers of reps..
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