Ch. 4, Probability and Sampling Distributions.
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:
Population
Choose from it a Sample
(varies)
Calculate
Numerical summary: Parameter(Greek
letter) Statistic
(Latin)
Example:
Population mean mu
Sample mean xbar
etc.
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.
First
piece of the pattern:
Law of Large Numbers (p.237) Take observations at random from a population with population mean mu. Then as the number of observations n increases, the sample mean xbar gets closer and closer to mu. (Even if the population is infinite! Note--we don't say how big n needs to be for how close here.) This won't be on the exam.
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 of the phenomenon.
Spinning
penny: Proportion of heads p =
(??)
Fuzzy word: Random numbers
are each equally likely. A random phenomenon
doesn't necessarily have outcomes
that are equally likely--all that is needed is that their relative
frequencies settle down to something.
Probability rules: pp. 222-3, in
words, then in notation.
A an event in sample space S, P(A)
is "the probability of A"
These rules are all true for
proportions in long run, 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:
Assign a probability to each outcome (>0)
so they add to 1. (Sometimes equal values 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 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(2Heads) = ?
Sample space | Y |
N |
Prob's
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.25| .75 |
Often the sample space is naturally expressed in numbers, thus
Random Variable: (X, Y, Z..) Variable
whose value is a numerical outcome of a random phenomenon.
Probability distribution of X tells
us what values X can take and how to assign probabilities to them.
If X has a finite number of
possible values, nothing new except notation.
Flip coin twice. R.V. X = number of heads:
Distribution given by table.
x| 2 | 1 | 0 |
P(X=x)
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.25| .50 | .25| P(X > 1)= ?
P(X = 2) = ?
HW Day22, Monday, Oct. 22 (Re)Read 4.2, to p. 27, 4.3, pp.236-38. Next: Read rest of 4.2, 4.3. Skip 4.4 & Ch. 5.
| Hand in:
Sec. 4.2 Probability models: p. 221 4.14 sample spaces p. 224 4.16 social mobility in Denmark 4.17 cause of death 4.18 husbands' share Finite sample spaces p. 226 4.19 legitimate dice? 4.21 p. 232 4.31 SRS size 2 4.32 farm size Random variable language--finite sample spaces still p. 231 4.25 sum of 2 dice p. 235 4.35 social mobility in England = = = = = = = = = Section 4.3, thru p. 238 p. 238 4.38 law of large numbers |
Read, to discuss
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Optional
(more of same)
4.28 land in Canada
p. 235 4.36 class (R.V. language) |
| Sievers home | Math151-Fall01/DayS22.htm | 11pm | 10/21/01 |