Math 151 , Fall 2005,  Day 23 Wednesday Oct 19 Hit reload ...After class, modified

Exam 2 Monday (Day 25, Oct.24).  Covers thru Today's HW, but no more than part III . Sign up today or Friday if you need a special time to take the exam.
How much computational detail from part II?  You don't need to know the formula for the correlation coefficient, but you should be able to guess roughly the r from a scatterplot, and know and use the properties pp.121-2.You will need to know, among other things,  how to find b₀ and b₁ from the means, standard deviations, and r of the x-and y-values,  and to give the formula for the regression line, (like 17, p.154); and to graph the regression line on top of the scatterplot.  Also find by hand the value that the line predicts for a particular x.  You should be able to identify and calculate the residual value for a particular x-y point as its vertical distance from the line (negative if the point is below the line), and identify and understand potential influential points.  You should know  that the regression line goes through the point given by the two means, and that the  regression line "rises" r standard deviations in y for each standard deviation increase in x (pp. 137-8); also that the regression line of "weight" on "height" is not the same line as the regression line of "height" on "weight" . You should be able to describe verbally the meaning of R² in the context of a data set.

Day 23 (Wed. Oct.19): Finish: D&V Ch 12, 13. Review part III p. 262.  AS13.  Bring exam questions...
    Next, D&V Part IV: Ch. 14, Ch.15 thru p.  291 (then Ch. 18 &on.) ActivStats is very good for part IV--Ch11"Randomness" shows Law of Large Numbers as D&V express it. Ch14, 15 "Intuitive Probability"&"Probability Rules" correspond well with the text and present very good examples. (You don't need SPSS for any of this...)

Hand in  Chapter 13, p257ff. 1,2,4,5,6,10,11,12, 17, 18  Finish these for those that are experiments 32 d Shingles, "better" design  35 Safety switch 36 Washing clothes From Review part III, p. 263ff. 26 Laundry 34 Pubs If you haven't handed it in already: Hand in answers to these questions on the "Placebo Effect" articles (outside my door/on  reserve) Hand in WEDNESDAY:  FRIDAY  a) Give two examples of the placebo effect (from the article!)   b) What do researchers believe causes the placebo effect?    c) In the separate article: "Pill will make you feel better...," what country was surveyed? Postpone the rest= = = = = = = = = = = = = =  ActivStats, Ch.11 HW, ACT 1 and ACT 2: (The disk in your book is fine for this; you don't need SPSS. ) In Ch. 11("Understanding Randomness") click on the "house" icon on the top menu bar to get the HomeWork.  Do problems ACT 1 and ACT2, using the "Randomness tool" which opens when you click on the button in the HW problem. Chapter 14, p. 280ff 1 Roulette 3  Winter 6  Crash 9 Spinner 11 a Car repairs 13 a M&M's

Read,   to  discuss    Review Part III:  p. 263 ff: 1 thru  17 odds, +12, 18      = = = = =

Op- tion- al

Homework questions? Day 22
Questions for exam?
Chapter 13: Experiment: Continue block designs Day 22   Brief summary:   All about avoiding BIAS
Principles of designing a comparative experiment (p. 243)

• Control: Control sources of variation other than our factors, by making everthing else exactly the same. Use specific levels of the factor, same within each group. (unfortunately, limits generalizing to other levels not used)
• Randomize the assignment of units to treatments: reduces possibility of confounding some characteristics of exp. units with the treatments.
• Replicate:  In the single experiment:  replicate the treatments on many exp. units:  Reduce the chance of unrepresentative results, assess the variation within a treatment.

   (Replicate the entire experiment in another time, place, setting, on another group.)
 
• (Block--not required:  Like stratifying--break into like blocks (M/F, age groups), then do above within each block. )

Exam 2 ends here
Start here next = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Part IV: Randomness and Probability.  (Why?)

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.  Need probability.
Recall (Day19):p. 227      SampleChosen from a  Population
  Numerical summary: Statistic (Latin)  Parameter(Greek letter)

The actual value of the Statistic will vary, depending on the particular sample. "Sampling variability" = "Sampling error"
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"  (Old stats text.   Caveat: rounded?)
                                        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
    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)

Chance  behavior (a random phenomenon): Unpredictable in the short run,  predictable regular pattern in the long run.
(Prof. Persi Diaconis (a table magician) can flip a coin so precisely it always comes up the way he wants.His coinflipping is not a random phenomenon.  Mine is.
"Probability" of particular something happening: proportion of times it would happen in a very long series of independent repetitions (trials) of the phenomenon: "long-run relative frequency".
    (independence:  outcome of one trial must not influence the outcome of any other.)

Law of Large Numbers (LLN):  Relative frequency of repeated independent trials gets closer  to the "true" relative frequency as the number of trials increases.
  (But it may take a long time: Large Numbers of trials. Use  http://www.whfreeman.com/scc -- "Probability " 1 toss at a time--settles down slowly.   )
(&&Another version of  LLN says the mean from a sample of size n gets closer and closer to the true = "population" mean, as you take bigger samples (as n increases).  Activstats presents this, 14-1, and we'll return to this soon.)

Aberrations won't be compensated for; they will only be swamped out.  (Misconception of "law of averages.")

Probability Model:
A Random phenomenon,
    Sample space S:  set of all possible outcomes (no overlap of descriptions) (def. p. 284)
    Event:  any  set of outcomes(including one outcome, & even the set containing no 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.
    S₁ = {HH, HT, TH, TT}     S₂ = {0, 1, 2} number of heads   S₃ = {Y, N} both are heads?

Probability rules:  (pp. 274-6, 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), prop.of counts, proportions of areas.
    1.  0 < P(A) < 1
    2. P(S) = 1
    3. For any event A, P(A does not occur) = 1 - P(A)
    4.  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)

Pick one person from U.S. Pop. (Age 25 +)

Sample space:	No HS degree	HS only     .	1-3 yrs College	4 + yrs College
Proportion in pop.	18.3%	33.9%	24.8%	23.0%
Probability	.183	.339	.248	.230

P(No 4-year "degree") = ?
P( HS or less) = ?

Finite sample spaces (you can list the outcomes):
Assign a probability to each outcome (>0) so they add to 1.   (Sometimes equal values--"equally likely" make sense.)
    Prob. of an event is sum of prob's of its outcomes.

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      |