| 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 (I just read that 1/5 of the military dying in Iraq are on their third tour (year) of Middle East duty) 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")
|
Start here 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.
Example: "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)
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?
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. (Or sometimes its logarithm will be normal instead--the
"lognormal" distribution) (pp. 366-7)
| Sievers home | Math251-Fall05/Dayps27.htm | 11am | 10/2805 |
