| Hand in:
Sec. 3.4 p. 240ff 3.71 b,c (sample means for Workers) This file is not in SPSS form, though it is in Text form (Appendix folder). Skip part a. 3.73 Applet coin flips. Take 5 samples, not 50, for each of parts a and b, and we'll pool them next time. Do a dotplot instead of a histogram since you have so few points. Review: p. 247ff |
Read, discuss
3.69 states p. 247ff - - - - - - - - - |
Optional - - - - - - - - More of the same: 4.10, 4.16, 4.17, 4.18 Blood type inheritance, 4.36-39 |
3.4, Toward Statistical
Inference. Details Day 18
Outline: Day 19
3.4 conclusion
Note, Sample size determines variability of sampling
distribution: Population size doesn't matter p.238 (as long as
pop. is at least "100" times the sample size (even at 10 times, you
still aren't getting enough of the population. to have much
effect.) A spoonful of soup gives you just as good a sample,
whether from a small pot or a caldroun. A toothpickful of soup is
just as bad...
Recall: Random number distributions have more
variability than you might expect.
Pickadigit.htm
Sample means and proportions are also variable.
We'll be able to put numbers on this variability soon.
Pooled data:
#3.77 Means from samples of size 20, CSDATA GPA means data, GPAmeans histogram
Results = part of Sampling distribution of the
statistic
which is used to estimate the parameter
µ.
Chapter 4: Probability
Sec. 4.1
Random phenomenon= Chance mechanism. Any individual outcome is
uncertain,
BUT in a large number of repetitions, a regular distribution of
outcomes
occurs.
Variability happens. In the long run it settles down.
Probability of any outcome of a random phenomenon:
proportion of times it would happen
in
a very long series of independent
repetitions (trials) of the
phenomenon: Long-term relative frequency.
(This is the "frequentist" interpretation. Mainstream)
(independence:
outcome of one trial must not influence the outcome of any other.)
http://www.whfreeman.com/ips5e/,
Probability applet
A Random phenomenon:
Sample space S= Set of all possible outcomes. (List or
description, no overlaps )
Event = outcome or set of outcomes (subset of S)
Probability model: S,
and a way of assigning a probability to each event.
Probability rules (axioms): A
an event
in sample space S, P(A)
is "the probability
that A occurs"
p. 262
1. 0 <
P(A) < 1
2. P(S) = 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) = P(Ac) = 1 - P(A)
Ac = "A complement"
ANY method
of assigning probabilities to sets must result in a system that obeys
these.
Note: All the following obey Rules 1 thru 4: All
can be used as metaphors or models for probability models.
Area
(If area of whole space is 1) Venn diagrams of Axioms
Volume (If
volume of whole space is
1)
Weight
(If
weight of whole space is 1)
Proportion
(count
of items in set/ total count)
Equally likely probabilities: If there are k
possible
outcomes, and each is equally likely, each has probability 1/k (Dice,
coin,
etc. Plenty of things are not equally likely.)
Pick one person from U.S. Pop. (Age 25
+) Note how we
can change a population distribution into a
probability mechanism by saying "choose an
individual at random from the population. " (problem 4.17--different pop.)
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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
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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
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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
\second
first H T
H | HH HT
T | TH TT
Rule 5. Multiplication rule for Independent events.
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.
Independence (rule 5: Multiplication rule)
is used two ways :
(a) When building a complex probability
model from simpler ones, using the mult. rule to assign probabilities
.
(b) Checking if two events
are independent in a known model, by seeing if the mult. rule
holds.
| population of 1000 |
Women | Men | Total |
| In Labor Force | 350 | 450 | 800 |
| Not in Labor Force | 150 | 50 | 200 |
| Total | 500 | 500 | 1000 |
Statistics question: if the 1000 is a sample from a larger population, is the difference from independence we see great enough to exclude the possibility that Women/In L.F. are independent in the population; or is the difference just the result of sampling variability?
| Sievers home | Math251-Fall07/Day2s20.htm | 11am | 10/10/07 |