Math 151 , Spring '08 Monday Day 25, March 31Hit reload...after Day 26 class.

HW:  Finish Chapter 9.Check p. 228: 9.16, 17, 18, 20 (obs/expt, factors)  Then 21 (choosing groups)(they should be done already), then 9.19, 22, 23, 25 (types).   Read Data Ethics, pp. 235-242 if you haven't. Read ahead   Ch. 10 to p. 256, def. of Random Variable (discrete) p. 260.  Check p. 263ff. 10.19, 20, 22, 23, 24, 25, 26, 27.
Hand in  Wednesday; bring questions for exam
p. 226, 9.13 hand strength, MP
p. 231, 9.35 forest CO2

p. 226, 9.15 teaching techn.  Why might I call this a  matched pairs rather than a general block design?   Don't actually do the randomization, but think about what ought to be done; we'll talk about it.
p. 232, 9.40 TV ads, block design.  Use the  Applet, to assign your subjects.  Number your Women and your Men, and show their numbers as well as the group they're in. 
p. 229, 232, 9.27 and 9.39 wine, beer, spirits two ways
- - - - - - - - - -
Hand in Monday:  "Ethics": Read Data Ethics, pp 235-242.  Find at least one other person in the class, and together discuss  one of these questions.  Write up your answers (If you have consensus, fine! If you disagree, say who thinks what).  pp. 242-245, # 4 or 5 or 9 or 11 or 13 or 14 or 17
= = Chapter 10: how much tonight??(NONE; not on exam).. =
p. 249, 10.1 Texas Hold'em

p. 265, 10.30 Sample spaces, free throws
p. 252, 10.5 Sample spaces
..
p. 249, 10.3  50, 200 Random digits.  Bring your result for (b) to class to compare with others.  I did it twice, got .06, .09   Applet:  Probability
p. 250, 10.4 Probability says..
 Read, to discuss

p. 232, 9.38  spine fractures You lack the information to make a complete design (i.e. how many women at each hospital.)  Sketch in what you can.

 Optional 


p. 226, 9.14 matched and not, more practice
Exam this Friday APRIL 4 (Day 27).  Bring one sheet of notes.  Chapter 5 (Material on assignments, Day 19 on: reading p. 120, p.123 ff(facts 2,3,4), p. 126-37).  Ch.7 p.171 G5-8. Chapters 8 and 9, all --through HW assigned today (but not Ch. 10).
   Sample exam available today in class, or in white folder outside my door.  Solutions Here.Note that there is no problem on the sample involving two (or more)" factors" or expressly listing "factors" and "treatments", but such questions could be on the exam. 
I have been known to ask questions on the exam specifically on the "outside" reading, such as the Placebo Effect articles.
Review Normal distribution: we'll want it soon after the exam. 
Homework questions:  Day 23
Placebo effect:  b) What do researchers believe causes the placebo effect? 

Ch. 9 Designing Experiments, Summary. See Day 22 ,  Day 23  for more  notes

Principles of designing an experiment: Compare groups with different treatments:   Control as much as you can, to make all the groups the same except for treatments, Randomize the rest; Use enough subjects  to average out bad "chance" .
   "Randomized comparative experiment"

More issues:
--Placebo and biasing effects--avoiding:   "Blind",  "Double blind."
--Lack of realism:   Do sociology, psychology experiments generalize to "real life?"
--Subjects are not a random sample from the population. ( Ivy League males, before 1970's.)
--Ethical questions...Milgram. Zimbardo prison. Whole section BPS4e, pp. 235-242

  = = = = = = = = = = = = = = = = = = = new today:  See Day 23 for details = = = = = = =
Statistical Significance p.221: An observed effect so large that it would rarely occur by chance (assuming no real difference in treatments) is called "statistically significant".   "So large", "rarely", "by chance" will be defined and quantified in Ch. 6.

Completely randomized experiment: all subjects are allocated at random among the treatments.
Fancier Experimental designs (not "completely randomized") Control extraneous variability by pre-sorting individuals into  homogeneous groups.  (BPS4e pp. 224-226)
Matched pairs: To compare Control and experimental treatments (i.e. 2 levels)
   Sort experimental units into "matching" pairs.   One member of pair gets control, other gets experimental.  Randomize which.
        Compare within pair, then summarize all comparisons.   Often: matched with self="self-paired"). (Randomize order)
Block design:  Sort experimental units into "Blocks" = groups homogeneous on potentially confounding variables
        Within each block, randomize the treatments. Compare results  within each block, then summarize all results.
    (Matched pairs is a special case of block design--each pair is a "block".)   Diagram p. 226

+ + + + + + + +new--finishing Producing Data section: + + + + + + + + + + + + +
 In practice, you may not be able to do the ideal experiment:  Sometimes the treatments cannot be deliberately  imposed (ethical reasons, practical reasons) and we must observe the explanatory variables (and the response) . (Can't force people to smoke.)   Included in this may be even intrusive measurements, assessments.
Not in text: "Prospective study--retrospective study."  (Both  are observational studies)
Observational Study:  Observe individuals; don't do anything to them; do not influence the responses.  Can indicate strength of relationship, differences, but not cause and effect.  (Often not with samples, but with selected group(s).)  Lurking variables?!? (Fisher:  Smokers smoke to soothe irritabilities that may cause cancer.)
----Retrospective:  gather data after the fact  (Observe that x% of men hospitalized with heart disease were/are smokers.  ) Some of the problems:  reliability of memory, completeness and reliability of records.  Comparisons are hard.  Others... 
March 14, '07: NYTimes and others: 12.9% of people admitted to the hospital with heart attacks on the weekend are dead a month later.  12% of those admitted during the week are dead a month later.  (Based on records of NJ hospitals 1999-2002).  A significant difference because so many people were involved (probably 13,000 or more).  Weekend people were much less likely to have angioplasty.  But a weekend/weekday difference was apparent even before angioplasty existed.  And they didn't track how sick people were coming in.
 ----Prospective:  choose individuals in advance. Measure them; and follow them, as events happen.  Still have problems of lurking/confounding variables. (Framingham Heart Study: 5,209 (2,873 women and 2,336 men) healthy residents between 30 and 60 years of age.  Followed from 1948 to now. A second-generation cohort recruited 1971, Minority group 1995  http://www.framingham.com/heart/)
Got to here Monday.
recall:Toward Inference: Table p. 186, Exploratory Data Analysis vs. Statistical Inference
Chapter 10, Probability (intro)
Chance  behavior (a random phenomenon): Unpredictable in the short run,  predictable regular pattern in the long run.
  (Random numbers:  equally likely in the long run.  "Random" in this chapter  is more general--pattern is not necessarily equally likely)

"Probability" of particular something happening: proportion of times it would happen in a very long series of (independent) repetitions of the phenomenon. ..  Applet:  Probability
    (independence:  outcome of one trial (repetition) must not influence the outcome of any other.)
The random number table.  At each place, the probability of any particular one of the 10 digits is 1/10, or .10.     Sets of 25 digits  from the table. Individual sets of 25 showed much variability.  Pooled  shows more "flatness" --but still much variability.  You would be right to be skeptical when I told you that your "pick-a-number" choices were not random, on the basis of just this class's data.  Not enough to necessarily show the pattern.

Probability Models : (p. 250-256)
Random phenomenon, described by
    Sample space S:  set of all possible outcomes (no overlap of descriptions)
    Event:  any outcome or set of outcomes
    Probability model: S, and a way of assigning a probability to each event.
Sample space depends on what you want to know:
Phenomenon: Flip coin twice.   3 possible sample spaces:
    S1 = {HH, HT, TH, TT}     S2 = {0, 1, 2} number of heads   S3 = {Y, N} both are heads?..

Start here Mon. Day 28
Probability rules:  pp. 253, in words, then in notation.
A an event in sample space S, P(A) is "the probability that  A occurs"
    These rules are all true for proportions in long run (Probabilities), proportion of counts, proportions of areas.
    1.  0 < P(A) < 1   (any probability is a number between 0 and 1. )
    2. P(S) = 1         (all the outcomes together have total probability 1)
    3.  A and B are  disjoint if they have no outcomes in common (can't happen simultaneously.)
        If A and B are disjoint, their probabilities add:  P(A or B) = P(A) + P(B)
    4. For any event A, P(A does not occur) = 1 - P(A)

Pick one person at random from U.S. Pop. (Age 25 +)   Probability = proportion in the population
Sample space:
No HS degree
       HS only     .
1-3 yrs College
 4 + yrs College
Proportion in population.
18.3%
33.9%
24.8%
23.0%
Probability 
.183
.339
.248
.230
P( that the person's education is HS or less) = ?
P(Not finished college or didn't start) = ?

Discrete models: (Can make a list of all members of the sample space)  Make the list, and
Assign a probability to each outcome (>0) so they add to 1.   (Sometimes equal values make sense.)
    Prob. of an event is sum of prob's of its outcomes.

Recall exercise 8.10:  Pick 6 people from list of 28 managers.  How many people of East Asian surname do you get?  Excel analysis

Sample space :
 0
 1
 2
 3
 4
 5
 6
Probability (from theory)
 .268
 .447
 .235
 .047
 .003
 .000
 0: impossible
Proportion out of  19 usable HW's
 .211
 .474
 .263
 .053
 .000
 .000
this class --significantly different?
 .727 (8)
 .182 (2)
 .091 (1)



(You won't learn how to calculate these probabilities.  Ch. 13, which we'll skip, gives a hint. )  chisq = 17.6
The chance of our seeing another 11 random samples that are this far "off" from the theoretical proportions, if they really were random samples, is less than 1 in a thousand.  This would be "statistically significant" evidence that some of the 11 people dry-labbed it--but since I wasn't doing an experiment deliberately to check on that,  I have to keep in mind that maybe this just was that 1-in-a-thousand event.  Mythbusters episode 50, poison oak vs. vodka.  It took 6 tries to find a person not immune to poison oak, though only 10 % of the population is immune.  Chances of that are .00001, 1 in 10,000.  Unusual things DO HAPPEN.  On the other hand, people may react only on 2nd or 3rd exposure, so 10% may be wrong for never exposed population.

Phenomenon: Flip coin twice.
    S1 = {HH, HT, TH, TT}     S2 = {0, 1, 2} number of heads   S3 = {Y, N} both are heads?
Sample space  | HH | HT | TH | TT |
       Prob's | .25| .25| .25| .25|  P(tail followed by head)=?
Sample space  | 2  |    1    |  0 |  P(at least 1 tail)=?   P(1 of each) = ?
       Prob's | .25|   .50   | .25|  P(at least 1 Head)= ?  P(2 Heads) = ?
Sample space  | Y  |       N      |
       Prob's | .25|     .75      |


Often the sample space is naturally expressed in numbers, thus
Random Variable:  (X, Y, Z...) Variable whose value is a numerical outcome of a random phenomenon.
 Probability distribution of X tells us what values X can take and how to assign probabilities to them.
    If X has a finite number of possible values (Discrete distributions), nothing new except notation.
     P(X < 2) is "Prob. that X is less than 2."
Flip coin twice. R.V. X = number of heads:  Distribution given by table.
             x| 2  |    1    |  0 |
       P(X=x) | .25|   .50   | .25|  P(X > 1) = ?   Words:  Prob that # heads is > 1
                                     P(X = 2) = ?         Prob that # heads is 2

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