Exam 2 next class (Day 24, Apr.1). Covers
thru Day 22 HW (Not Block or Matched Pair designs).
Sign
in today 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. March 30): Finish: D&V Ch 12, 13. Review part
III p. 262. AS13.
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.
| Hand in Postpone
all till after Monday's lecture
Chapter 13, p257ff. 1,2,4,5,6,10,11,12 You did the "observational study" ones, and started the "experiment" ones. Finish these for those that are more complex experiments, add 18 32 Shingles part d 35 Safety switch = = = = = = = = = = = = = =
9 Spinner
Using independence:
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to discuss Review Part III:
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Homework questions? Day
21
Day 22
Questions for exam? Took
whole class, good discussion. Start here Monday
Chapter 13: Experiment: Continue Day
21 Brief summary: All about avoiding
BIAS
Principles of designing a comparative experiment
(p. 243)
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. (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
Sample
Chosen 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"
(Moore stats 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
Fall '04, xbar = 64.68, s = 3.54
Spring '05, xbar =64.31 ,
s =2.93
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 +)
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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
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Prob's
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.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
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.25| .50 | .25| P(at least 1 Head)=
? P(2 Heads) = ?
Sample space | Y |
N |
Prob's
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.25| .75 |
Flipping-coin-twice was built from a simpler phenomenon; flipping coin once: P(H) = .5, P(T) = .5
Rule 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.
Rule 5 can be used to build probabilities
for complex phenomena from simpler ones (Ch. 14); to check structure
in existing sample space (Ch. 15.)
e.g. Pick 2 people at random from U.S. pop. (Pop. is so
big that it's hardly changed by removing first. Independence OK)
P(First has 4+ yrs college, and 2nd didn't graduate HS)
= .280×.183 = .051
P(First didn't graduate HS, and 2nd has 4+ yrs college)
= .183×.280 = .051
P(one didn't graduate HS, and the other has 4+ yrs college)
= .051+.051= .102
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