Math 151 , Day 27, Friday, April 9 2004 hit
reload...After class
HW Day27 Read sec. 4.3. (Skip ch. 5)
Sec.
6.1 next. Read ahead...
Memorize the tan box on p. 242 (mean and
s.d. of sampling dist. of X-bar)
Moore sec. 4.3
Hand in Wednesday: Work on them and come with questions Monday.
DIST. OF XBAR(S)
These problems use only the mean and standard deviation.
p. 243, 4.41 (lab measurements)
p. 250, 4.50
These problems use either the Central Limit theorem, or the "sample
mean of n independent observations from a normal distribution has a normal
distribution." theorem (both on p. 244)
p. 249, 4.51 cola (you did a, now do b)
p. 247, 4.44 carpet flaws. Also draw some square
yards and mark some flaws.
p. 250, 4.53 auto accidents
More problems:
p. 243 4.42 unbiasedness, sample size
p. 249 4.52 hypokalemia
p. 249 4.48 dust Note, the dust actually weighs 123mg,
but the weighings may not be accurate enough for us to find the actual
weight. "Distribution of this mean" = "Distribution of means from all possible
sets of 3 weighings from these scales." When I took physics,
we did not have digital scales; they were balance beams; and we weighed
everything 3 times and found the average. (Have you ever gotten on the
scale, said "that can't be right!" gotten off and on again a couple times?)
.
p.250 4.54 (labeled 4.53?) pollutants; backward
from value to probability. You might want to know L so that if
you tested your 125 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 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 hit that 1 in 100 possibility." |
Read,
to discuss |
Optional
Extra credit,
see LLN-game
Due Monday |
Exams not finished. Sorry! Monday for sure!
- HW questions? Continuous Random Variables. Normal Random Variables.
-From # 4.40 p.241 Add
your 3 means to the circulating list.
= = = = = = = = = = = = = = = = = = = = = = =
How does sample mean behave? (4.3)
Sample Chosen
from a Population
(varies)
(fixed, but usually unknown)
Calculate Numerical summary:
Statistic
(Latin)
Parameter(Greek
letter)
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.
Activstats p 15-3 activities.
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?
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" (Sec. 4.3)
(sampling dist. of (sample) proportion: Spinning
penny )
Shape, center,
spread, (outliers?)
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.
Look at results from #4.40.
Demo Activstats
18-2-1 (lln), 2(s.d.) 18-3-1(clth)
Things we know:
-
Whatever the population distribution of X,
that we draw the sample from, (see p. 242)
The mean of the x-bars = the mean of
the population
The standard deviation of the x-bars =
the s. d. of the population divided
by the square root of n.
-
Consequences:
* X-bar "hits" the population mean on average--is
"unbiased estimator" of µ (doesn't systematically go too high
or too low.)
* Averages are less variable than individual
observations. Averages from large samples are less variable than averages
from smaller samples (because of dividing by the square root of n)
-
IF the population is Normal, the
sampling distribution of Xbar is Normal.
-
"The Central Limit Theorem
In any case, for "large" n, the
sampling distribution of Xbar is Approximately Normal.
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?
Take SRS
of 9 people. Sampling distribution of the mean? Probability
that the mean is below 98.0?
Probability that one (random)
healthy individual's normal temperature is above 98.8?
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 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)
Got roughly to here, but will review, then
go on.
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.
Even if the population is
pretty weird, n=25 gives a pretty good approximation to normal.
Pictures on overhead.
This page belongs to Sally Sievers who is solely responsible
for its content. Please see our statement
of responsibility.
