Math 151 , Fall 2004, Monday Day 22, March 29 After class Hit reload...

HW assignment Day 22  Bring questions for exam
Reading: ReRead section  3.2, Significance, Read Matched pairs and block design; review ch. 3. Review ch.2
    Next:  4.1, 2, 3.  We'll do  4.1, 2, 3.  Skip 4.4 and Skip Ch. 5.

Hand in: Matched pairs and blocks  p. 199 3.43 hand strength 3.45 weight loss 3.44 student traders.  The difference in the treatments is whether or not they have software that can make "charts" of past "trends." (If they don't have the software that "highlights trends" they don't have "charts"--they just have lists of numbers giving the price history.) = = = = = = = =  Probability: Sec.  4.1  p. 215,  4.1, 2, 3 parameter/statistic Postpone:  4.9  3 of a kind  4.10 numbers-->words  4.12 world series prob?

Read, to discuss  p. 209 3.72 McDonald's vs Wendy's  p. 209 3.71speeding the mail   = = = = = =  Probability: Sec.  4.1  Postpone p. 218 4.6 random digits

Optional  (more of same)  p. 203, 3.58  3.59   = = = = = =

Exam 2 Friday (Day 24)  Covers Chapters 2 and 3
Sample exam problems: "Sample exam 2" given out: (Copies), Solutions outside my door+ on reserve.
How much technical detail from sec. 2.2 and 3?  see Day 21, also in 3.1, Stratified, Multistage, and Systematic samples not required.

HW questions, Designing experiments? see Day 20
        "Control, randomize, replicate"

Fancier Experimental designs (not "completely randomized") Control extraneous variability by presorting individuals into  homogeneous groups.
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.
  Common: Do the control and experiment to same individual (matched with self). (Randomize order)
        Are right feet bigger than left feet? (not an experiment)      Sunburn salve experiment?
    Aside:  Sampling data, "longitudinal study" following same people through time.
            Works like matched pair to control variability.
Block design:  Sort experimental units into "Blocks" = groups homogeneous on potentially confounding variables
     e.g. M/F, age, income, weight, fruitflies wild or curly-winged.
    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".)

Not in text:  In practice, the ideal requirements may not be met:  Sometimes the treatment cannot be deliberately  imposed and we must observe it (and the response) when it happens. (Can't force people to smoke.)
"Prospective study--retrospective study."
--Prospective:  You get your subjects before something  (e.g. disease) happens to them, can get information from them.  Then it happens (or doesn't).  E.g. enlist 1000 women, collect info, wait 5 years.  See who gets the disease. An observational study, but More like an experiment than
--Retrospective:  Ask people with/without disease what they were/are like.  (Problems: Reliability of remembered info,  matching,  sampling)  (My mother's headaches)

Ch. 4, Probability and Sampling Distributions.
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)
25 digits from the random number 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.
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
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.

Sec. 4.1:
         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
Start here next time:
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"  (text p.66.  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
    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)
   Spinning penny From p. 216, #4.4, HW Day 21  :
         Each of you took a sample of size 50 from the population of all possible penny spins, got a statistic, p-hat.
        Add YOUR result to the list going around.  Also tell whether you FLICKED or TWIRLED the penny.
     Results so far from spinning penny
' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '
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" here is more general--pattern is not necessarily equally likely

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 of the phenomenon.
    (independence:  outcome of one trial (repetition) must not influence the outcome of any other.)