Math 151 , Day 29, Wednesday, April 9, 2008 hit reload....After class

Exams not finished.  Sorry again!
Hand in now, your sheet of results from 10.55 and 10.56 (Probability applet results)  If you did 10.3 since Monday, add your results to the sheet circulating.
- HW questions?Day 28  Discrete, Continuous Random Variables.  Normal Random Variables.  Notes Discrete Day 25 Continuous Day 28 
.Wed: We went through sample/population and statistic/parameter, and Law of Large Numbers.
I introduced distribution of Xbars for fixed sample size n, stated the "big facts" : Expansion
    Mean of Xbars' distribution = mean of population.
    S.d. of Xbars' distribution = (s.d. of population) divided by (square root of n)
      If population is Normal, distribution of Xbars is Normal.
      If population is NOT Normal, distribution of Xbars is still approx. Normal, if n is big enough.

Examples and details next time, the computer gods willing...

<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 
                                                                    Spring '08  xbar =65.04 ,    s =3.91
<>   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:.. 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...)
Start here Fri. with the Applet demonstration, we hope... then forward to big facts

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.

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.
  
•  Rice U. Applet  (do Sample: Animated sample; Distribution of means: n =5, then n = 25. Do Sample 5, 1000, 10000 after idea clear. )
• Normal X vs Xbar
  
• "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.) (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.)

...
  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..