(as long as the population is large compared to the sample (at least 10 to 20 times sample)
| Hand in Friday: If you didn't, Sampling experiments which were due today (10.55 and 56, 11.6, as modified. see day 28) Also: (you can do this one without much understanding of chapter 11:) p. 277, 11.6 sampling distribution of exam scores Do a and a modified version of b; Do b this way. Close your eyes and put your finger down somewhere on table B (Don't use row 116!! unless you land there.). Start reading the table where your fingertip lands. Record your sampleof 4, and 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). Record your 3 xbars, Make a dotplot of your 3 xbars and bring the values to class to be compiled with everyone else's.. = = = = = = = = = = = = = = = = = = 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 Nothing else new to hand in: PLEASE read Ch 11, read over the HW problems to get used to the words, questions, language here. This is THE BIG IDEA chapter for the remainder of the course! = = = = = Ch. 11--postpone= = = = p. 272 11.1 caffeine (Param./Stat.) p. 272 11.2 voters(Param./Stat.) p. 277, 11.6 sampling distribution of exam scores Do a and a modified version of b; Do b this way. Close your eyes and put your finger down somewhere on table B (Don't use row 116!! unless you land there.). Start reading the table where your fingertip lands. Record your sampleof 4, and 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). Record your 3 xbars, Make a dotplot of your 3 xbars and bring the values to class to be compiled with everyone else's.. p. 275, 11.4 means in action (LLN) p. 275, 11.5 insurance (LLN) DIST. OF XBAR(S) These problems use only the mean and standard deviation. p. 280, 11.7 (Teen cholesterol ) p. 280, 11.8 (lab measurements) For (b) they mean "what should n be?' These problems use the "sample mean of n independent observations from a normal distribution has a normal distribution." theorem (p. 278) p. 280, 11.9 NAEP math scores (n = 1, n = 4) p. 290, 11.37 and 11.39 Pollutants in auto exhausts For 11.39: You might want to know L so that if you tested your 25 cars and found a high value of x-bar, you would be able to compare it with L; if it was greater than L, you would go back to the manufacturer and say "I believe you sold me a batch of bad cars, because the chances of getting an average emission level this high if p. 289-90 11.36 and 11.38 Glucose testing If we use this cutoff level L to say that people (with a mean of 4 tests) over L "have diabetes", then the chances of declaring that someone "has diabetes" when they really are OK (with mean 125mg/dl) is .05. .05 or 5% is the chance of a "false positive" using this protocol, when the real mean is 125. These problems use the Central Limit theorem (p. 281) p. 185, 11.10 What does the CLTh say? p. 286 , 11. 12 SAT scores, n = 1 and 70 p. 286, 11.13, insurance (Hint: find P(Xbar> $275)) p. 298, 11.41 auto accidents p. 298, 11.42 airplane overloads (Hint: to do the problem you have to assume all the seats are taken. Maybe not a reasonable assumption, but if there are empty seats, there's likely not a problem with overweight.) the exhaust system is working properly is only 1 in 100. It is more reasonable to believe the exhaust system is not working, than that we "are" that 1 in 100 possibility." |
Read, to discuss |
Optional More practice on Normal: p. 261, 10.17, ACT scores - - - - p. 272,11.3 Bearings (Param./Stat.) |
Next: How does sample mean behave?
( pp.275-86)
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 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...)
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 --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.
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 infinite!
Note--we don't say how big n needs to be for
how
close here.)
Now: keep a fixed
sample size n:
What is the distribution of the random variable Xbar,
when the experiment is to take a simple random sample of size n? This
is the distribution of means of all
possible
SRS's of size n.
We'll call it the "sampling
distribution of the (sample) mean" (p. 275-7,
then details
278-86)
Whatever
the population
distribution of
X,
that we draw the sample from, (see p. 278)
(as long as the population is large compared to the sample
(at least 10 to 20 times sample)
SPSS
simulation: average of spinners
which can land on any number
between 0 and 1.
Population--one spinner. distribution flat 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
Xbars from SRS:
Mean of Xbars is mean of
population.
Standard deviation of Xbars is
s.d. of population divided by square root of n.
As sample size increases,
sampling
distribution of Xbars gets more and more normal-shaped.
(Central Limit Theorem)
Central Limit Theorem...
How large is "large"? How approximate is
"approximate"?
If the population was close
to normal, n doesn't need to be very large.
If the population is not badly
skewed or bimodal, n=25 already gives a pretty good
approximation to normal.
Author's website
applet, Central
Limit theorem for a highly skewed dist.
Pictures on overhead.
Applet
(kind of creaky) where you can change/create
the population dist. Sometimes it fails in pieces (sd=0)
or crashes if you try to use all the options, but is pretty good if you
stick to the Mean, and use only the top 3 displays, and don't go for
huge numbers of reps..
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