| Hand in: Friday
Prep for sec. 4.3: p. 241, 4.40 do a and b; Do b this way. Close your eyes and put your finger down somewhere on table B. Start reading the table where your fingertip lands. Find xbar for your sample. Now Repeat part b, to get a total of 3 values of xbar. (You can just keep reading the table where you left off, or you can put your finger in a different spot). Make a dotplot of your 3 values and bring the values to class to be compiled with everyone else's. Continuous sample spaces: (Should have done last time; hand in with
this asst.)
Normal distribution:
|
Read,
to discuss |
Optional
D. Law of Large numbers:
|
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.
(independence:
outcome of one trial (repetition) must not influence the outcome of any
other.)
http://www.whfreeman.com/scc
What is probability? 1 toss at a time--settles down slowly.
RECAP Sec. 4.2 Probability Models
Random phenomenon,
Sample space S: set
of all possible outcomes (no overlap of descriptions)
Event: any outcome
or set of outcomes
Probability model:
S, and a way of assigning a probability to each event.
Probability rules: pp. 222-3, 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)
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 & how to assign probabilities to them.
If X has a finite number of
possible values (Discrete distributions), nothing new except notation.
P(X < 2) is "Prob.
that X is less than 2."
Flip coin twice. R.V. X
= number of heads:
Distribution given by table.
x| 2
| 1 | 0 |
P(X=x) | .25| .50
| .25| P(X
>
1) = ?Words:Prob
that
#
headsis >
1
P(X
=
2) = ?
Prob that # heads
is
2
Random variables with intervals of outcomes ("continuous")
Sec.
4.2 pp. 228-232
If the sample space is an interval of values (or the whole line),
the way we assign probabilities to events is with a density curve (cf.
Sec.
1.3, Day
7) (remember density curves were idealizations of
histograms--of repeating the "experiment" many many times)
P(a < X < b) = the
probability that X is between a and b
is the area under the density curve, between a and b.
We declare P (X = a) = 0 , so P(a < X
< b) = P(a < X < b)
Notation: Use capital letter for the random
variable,
the "label" of the phenomenon. Use small
letters for particular values it can have. But this rule is
often broken--Moore uses x-bar where many would use X-bar.
B. Y = (number you get from) the sum of two spinners. ("Triangular")
a) The probability that the sum is a number less than .6 =
P( ?
) =.18
b) P(Y > 1.6) = ? P(Y < 1.6)
= ? P (Y <
1) = ?
P( 1 < Y < 1.6) = ?
c) P(Y > x) = .08. Find x: ?
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- - - - - -
Our most important probability model: NORMAL DISTRIBUTION family.
Same techniques as before, only we ask "probability that one chosen at
random..." instead of "proportion of all..." Review Normal techniques:
Day 8, Day 9
Take a random sample of size 1 from a population which is
N(110, 25). =
(Give an individual, chosen at random, the Wechsler test, which
has a normal distribution, mean 110, s.d. 25. X is the score
on the test.)
Find P(100 < X < 140), prob. that individual gets between
100 and 140. Work is on Day
9, what proportion.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
= = = = = = = = = = = = =
Next: How does sample mean behave? (4.3)
Sample Chosen
from a Population
(varies)
(fixed, but usually unknown)
Calculate Numerical summary:
Statistic
estimating
Parameter
xbar
µ
We take a simple random sample of size n, find the sample mean xbar.
It will be different depending on the sample, so we have a random phenomenon.
We measure the outcome as a number, the sample mean, so we have a
random variable X bar.
Law of Large Numbers (p.237, "LLN") Take observations at
random from a population with population mean µ.
Then as the number of observations n increases, the sample mean xbar
gets
closer and closer to µ.
(Even if the population is infinite! Note--we don't say how big n needs
to be for how close here.)
OR Let the sample size n get bigger. Then the xbars
will eventually get very close to the population mean µ.
OR As the sample size increases, the sample mean gets closer
to the population mean µ.
OR For a very large sample, the sample mean will (almost certainly)
be very close to the population mean.
e.g. the bigger my statistics class, the closer their mean height
should be to the U.S. mean height for women.
Now: keep a fixed sample size n:
What probability distribution describes the random phenomenon
of
finding
xbar from a SRS?
That is, what is the distribution of the random variable Xbar,
when the experiment is to take a simple random sample of size n? We'll
call it the "sampling distribution
of the sample mean" (Sec. 4.3)
This is the distribution of means of all
possible SRS's of size n.
(Spinning penny: sampling
dist. of sample proportion)
HW tonite #4.40: each get 3 SRS's of size 4, find 3 means:
will pool to get histogram of Sampling distribution.
Random number table: Let X = # of 0's in a set of 5.
Then each row is a sample of size 8. xbar = (sum of 0's)/8.
The 150 rows give us 150 xbar's. Look at
this approximation to the sampling distribution of the sample
mean.
Quincunx board: Result for one ball is "average" of going +
or going - at each level
(entry + pin 1+pin2+ ...+ pin 6).
What do we see?
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