Math 151 , Day 27, Friday, October 29 2004After
class
hit reload...
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)
Hand in sec.
4.3For Monday
LLN: p. 238, 4.39 betting on the
numbers (I'm told the numbers pays more on average
than
the NY Lottery)
Normal distribution review.
(NEW!)
A. Use the information on pp. 59-60 (examples 1.16 and 1.17)
to do the following:
a) Restate in probability notation and find the probability:
What is the probability that a 14-year-old boy chosen at random has
more
than 240 mg/dl of cholesterol?
b) Restate in probability notation and find the probability:
What is the probability that a 14-year-old boy chosen at random
has
cholesterol between 170 and 240 mg/dl?
B. Use the information from problems 1.88 and 1.90, pp.
76 and
77 to do these:
a) Restate in probability notation and find the probability:
What is the probability that an individual chosen at random from
the population used to develop the test has a score between 1.7
and
2.1?
b) A lottery with one big prize is to be held in the population
used to develop the test. Everyone has an equal chance of being
chosen.
In addition to the main prize, if the person chosen has an ARSMA score
in the top 30%, e will be given a trip to Yellowstone Park. (Thus
the probability of the chosen person winning the Yellowstone trip is
30%).
The organizers want to know the cutoff score for the top 30% of the
probability
distribution.
= = = = = = = = = = = = =
Postpone the rest:
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
|
Exams returned. Very good, overall! Comments
total #1 #2 #3 #4 #5 #6 #7 #8 #9 9|5
possible 100 |15 6 6 18 13 22 7 8
5 9|012234
max 95 |15 6 6 18 13
22 7 8 5 8|77
Q3 92 |14 6 6
18 13 22 7 7 4 8|34
Med 87 |13 6 6 17 12
20 7 5 3 7|
Q1 74 |10 6 6
16 11 16 5 4 3 7|14
min 61 | 4 3 1
8 7 8 0 0
1
6|133
Solutions on reserve (electronic, soon?) & outside my door.
60's or 70's, please see me!
= = = = = = = = = = = = = = = = = = = = = = =
- HW questions? Continuous Random Variables. Normal Random
Variables.
Day
26
-From # 4.40 p.241 Add
your 3 means to the circulating list.
= = = = = = = = = = = = = = = = = = = = = = =
How do sample means behave? (recap of
day 26 outline) (4.3)
Sample Chosen
from a Population
(varies)
(fixed, but usually unknown)
Calculate Numerical summary:
Statistic
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.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.
Start Here Monday
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 would give us 150 xbar's.
Look at this approximation to the sampling distribution
of the sample mean.
Look at results from #4.40.
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)
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. Author's website applet, Central
Limit theorem
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