Math 151 , Spring 2005,  Day 25 Monday April 4   Hit reload ...

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Chapter 13: Experiment: Continue Day 21   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. )

Block designs: (not "completely randomized")
(Randomized) 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 little "block":
Matched pairs: In experiment, 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 (find difference), then summarize all comparisons.
  Matched with self is common.  Eliminates extraneous variability.
     (Matching is also often used in observational studies)

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Part IV: Randomness and Probability.  Day 23  .    Highlights:

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.   If we choose simple random samples, we can understand the pattern of values the statistic can take.

Chance  behavior (a random phenomenon): Unpredictable in the short run,  predictable regular pattern in the long run.
"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. Aberrations won't be compensated for; they will only be swamped out.  (Misconception of "law of averages.")

A Random phenomenon,   Sample space S. ("Events")  Probability model: S, and a way of assigning a probability to each event.

Probability rules:  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)
 5.  If A and B are two independent events, the probability that both A and B occur  is the product of the probabilities of the two events.  P(A and B) = P(A)×P(B), if (and only if) A and B are independent.