| Hand in Wednesday .. = = = = = = = = = = = = = = = = = = READ Personal Probability, pp. 261-2. All our theory will be developed using the "frequentist" point of view (probability = proportion in the long run). But there is another theory based on Personal Probability, sometimes called "Bayesian". p. 262, 10.18 Continuous sample spaces: Normal distribution: Restate
each problem from "The probability that X is..." to "The proportion of
the population of x's that ...." and use your old techniques. Postpone the last 2 |
Read, to discuss
|
Optional |
Exams
not finished; sorry!
If you didn't Wed. and have it now: p.
249, 10.3 Please add your proportion of "heads" = "0's" from 200
repetitions ("tosses") to the list and dotplot circulating!
I did it 4 times (four sets of 200 tosses) , got .06, .09, .07, .14! The
list/dotplot
BTW, as you
watch the results Tues. night: "... [T]he primary value of exit
polling is to help us understand why people voted the way they did.
This is an entirely different task than trying to predict a winner for
Internet junkies who can’t wait a few more hours until actual votes are
counted. ...In every state, Republicans
are at least twice as likely as Democrats to say that they are not at
all willing to take an exit poll." Rasmussen
What kind of bias is that?
Summary:
Everything to here was Discrete sample spaces (you can
list the outcomes) New:
Random variables with intervals
of outcomes
("continuous") Ch.10 (p. 256 on)
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
(Ch. 3, cf. Day 6 on) (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)
Most important example for us; Normal distribution family
Details Day 26
= = = = = = = = = = = = = = = = = = = = =
Chapter 11, sampling distributons
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.
Ch. 11:
Sample Chosen
from a Population
(varies)
(fixed, but usually unknown)
Calculate
Numerical summary: Statistic (Latin)
Parameter(Greek letter)
Examples:
Sample mean xbar Population mean mu (µ)
Sample st. dev. s Pop. standard dev. sigma
Sample median
Pop. median
Sample proportion p-hat Pop. proportion p
Sample line height y-hat Pop. regression line height y
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.
Some examples of statistics:
Height: U.S. young
women: pop. mean= 64.5", pop. s.d.
2.5" (ed. 2 p.66.
BPS4e p. 76, 87 says women 20-29 is N(64, 2.7))
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
Fall '05 xbar =63.92 , s =2.80
Spring '06 xbar =62.93 ,
s =2.78
Fall '06 xbar =62.81 , s
= 2.65
Spring '07 xbar =65.18 , s =2.26
Fall '07 xbar =65.67 , s = 2.73
Spring '08 xbar =65.04 , s =3.91
Fall '08 xbar =64.89
, s =2.38
<> 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)
Got to here Monday
Next:.. How does sample mean behave? ( pp.275-86)
(=How do "all possible" sample meanS behave?)
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. 273-4,
"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 infinitely large! 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 the number of women in my statistics class, the
closer their mean height
should be to the U.S. mean height for women.
(Statistics means never
having to say you're certain... DID YOU
Watch the pollsters Tues. nite?
Applet:
http://www.whfreeman.com/bps4e
"Law of Large Numbers" Roll a single die. X = number.
µ=
3.5. (Think X is number of spaces you can move in a board game.
Average per roll is 3.5.)
My result at home --1st roll: x
= 5 n=1, Xbar = 5/1 = 5.
Roll again, x = 2. n=2, Xbar = (5+2)/2 = 3.5
Again, x = 1 n=3, Xbar =
(5+2+1)/3 = 2.67 Again...
... Xbar for large
n; close to 3.5.
| Sievers home | Math151-F08/Dayf28.htm | 12m | 11/4/08 |