Day 26 : Reading: Finish D&V Ch 13: . Review part III p. 262. AS 13. Ch. 14, and Ch. 15 thru p. 286 only. Then Ch. 18 &on. Start reading 18 now!) . 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 14, p. 280ff Using independence: (text below)
Chapter 15, p. 299
p. 352 #21 a, b only. Pregnancy Find the solutions, then Restate a and b from proportion questions to probability questions: "What is the probability that a pregnancy chosen at random will last... ... " and "If the chances of a random pregnancy lasting longer than k days is.... ..., then what is k?" p. 265 #28a only Rivets p. 353 #33a only IQ's, #34a only Milk |
Read,
to discuss Chapter 14
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Op-
tion- al |
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Part IV: Randomness and
Probability.
We want to use a Statistic
calculated from a Simple Random Sample to
estimate a Parameter of the
corresponding Population.
To talk about how good this estimate is, we must understand the Sampling
Variability (Sampling Error) of the statistic--how statistics
from different samples will vary.
To do that we need the language and concepts of probability.
Day
25 .
Continuing from Day 25:
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)
= .230×.183 = .042
P(First didn't graduate HS, and 2nd has 4+ yrs college)
= .183×.230 = .042
P(one didn't graduate HS, and the other has 4+ yrs
college)
= .042+.042= .084
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.
(Not explicitly in text:)
Continuous sample space: If the
sample
space is an interval of values (or the whole line), the
possible
outcomes are "x" or "y" values in the interval. The way we assign
probabilities to events is with a density (Day
8). (Remember density curves were
idealizations of
histograms--of repeating the "experiment" many many times.)
Area represents proportion-->> Area represents
probability.
P(a < x < b) = the probability
that the outcome x is between a and b
is the area under the model's density curve, between a and b.
is the proportion of x's which would come up between a
and b
if we did the phenomenon a zillion times.
We declare P(a) = 0 (In a continuous model, getting precisely
a
is utterly unlikely; can't even measure that well),
so P(a < x
< b) = P(a < x < b)
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