| Hand in: Binomial Normal approximation 5.17 alcohol study (this reasoning is that of significance tests, coming up) 5.23 proportion of sample 5.25 variability on exam Also: 5.14, 5.15, 5.16, Gallup poll: effect of n on margin of error 5.56 p. 376 common names (What are your chances if you're taking a statistics course? Lower...) Postpone? Sec. 5.2 5.29 axle diameter (mean, s.d.) 5.31 abc axle diameter (cf. X and Xbar) 5.35 mean number of moths in a trap 5.41 airplane weight (They suggest changing it into a "mean" problem, but you can do it as a "sum" problem using the algebra of means and variances and noticing that if Xbar is normal, so is the sum of the Xi's) 5.37, 5.39 Sheila's gestational diabetes 5.40 carpet flaws [5.39 and 5.40 both involve the computations needed for "significance testing". The remaining step is this: If our results are very unlikely, they call our assumption (the value of the mean) into question.] |
Read, discuss Postpone? Sec.5.2 5.33 unbiased 5.65 p. 377 dye pH (Read for intro to "control limits",
compare with
5.29 and 5.31. This is "Quality Control") |
Optional 5.62 p. 377 bomber tour of duty (In 2005 1/5 of the military dying in Iraq were on their third tour (year) of Middle East duty. No idea now...) Postpone? Sec.5.2 (more practice) 5.32 lightning strikes. (Use algebra of means and variances for part a) 5.61 p. 377 car passengers. Chaper 17, on your disk or downloadable, is all about
issues like
# 5.29, 31, 65--"Quality Control") |
.Repeat &
elaborate Monday.
5.2 (Central Limit Theorem and friends) generalizes the
idea of averaging (or summing) n independent identical random
variables
from the simplest case, where each is a Bernoulli trial S (outcome
either
0 or 1), to any X with any distribution. Nothing should
be
a big surprise. (Notational annoyance: In the Binomial
case,
we used X for the sum of the S's. Here X is the basic building
block
(like the Bernoulli trial S) and we'll sum the results of a bunch of
independent
copies of it.)
Xbar = (X1
+ X2 +...+ Xn )/n
,
where
the Xi 's are independent, identically distributed, all
like
the "population" distribution "X".
The X's are (usually for us) a SRS from a population with distribution
of X (assuming pop. is 20 times n)
The distribution of all
possible
Xbars is called the Sampling distribution of Xbar.
Note: They use capital letters for random
variables and small letters for specific values, except they
use
small xbar in this section where they should be using Xbar.
(Typesetting
problem??)
| µXbar = µX sigma Xbar = sigmaX ÷ sqrt(n) you know how to show these |
| If X is normal, then Xbar is normal believe |
| If n is large, then Xbar is approximately normal whatever X is (Central Limit Theorem) believe |
"n is large" --how large? Extremely skewed
(long-tail)
example; n = 80. Fairly contained range, even if somewhat skewed;
n = 25 Applet,
Central Limit Theorem. Some examples (overhead): sum of 3
dice.
Others.
Sampling distributions,
Central Limit Th. Rice
U. Applet
do
Begin button: In Sampling Distributions Applet, Sample:Animated button;
Distribution of
means: n =5. repeat Animated button.
After the idea is
clear, Do Sample: 5, 1000, 10000 after idea clear. Now
repeat with Distribution of means: n=25.
Try different "parent populations"--the top histogram
Example of computation: "Normal" body temperature
98.6 deg. on average. (Assume this is true.)
Assume normal distribution, & s.d.among many people
is 0.6.
Probability that one individual's normal
temperature is below 98.0 degrees? (z = -1) (Normal
Table)
Take SRS of 9 people.
Sampling distribution of the mean? Probability that the mean is
below 98.0? (z = -3)
Probability that one (random) healthy
individual's normal temperature is above 98.8? (z = .33)
Probability that the mean of a sample
of 4 is above 98.8?
Probability that the mean of a sample
of 36 is above 98.8?
Probability that the mean of a sample
of 100 is above 98.8?
Got to here Monday Day 28. Added visual aid: Normal parent and Xbar
SPSS simulation: average of
spinners
which
can land on any number between 0 and 1.
Population--one spinner. distribution flat
(uniform)
between 0 and 1, mean .49 s.d. = .29
n = 2, Average of 2 spinners is Xbar. Distribution
triangular between 0 and 1, mean .50, s.d. .21.
.29/sqrt(2)
=.205
n = 4, Average of 4 spinners is Xbar. Distribution
normalish between 0 and 1, mean .50, s.d. .15. .29/sqrt(4)
=.145
n = 15, Average of 15 spinners is Xbar. Distribution
normal between 0 and 1, mean .50, s.d. .09. .29/sqrt(15)
=.076
Why is the Normal distribution so common in the world?
"Fuzzy Central Limit Theorem" : Data whose variation is due to
many small independent random influences will have an approximately normal
distribution. (pp. 366-7) (Or sometimes its logarithm will be normal instead--the
"lognormal" distribution)
| Sievers home | Math251-Fall07/Day2s27.htm | 2pm | 10/29/07 |
