Math 151 , Day 28, Monday, April 8, 2002Hit
reload to get current version After
Class
SAMPLE from an UNKNOWN population.
Each person take 4 slips from the Birkenstock box,
find the mean, and the mean +
.841.
Record these for yourself . This
is your Confidence Interval Estimate of the mean of the Birkenstock
population.
Record them also on the sheet going
around, and draw the interval on the graph transparency going around.
Exams returned. Comments, and
makeup instructions.
If you missed class, you can get your exam from the box outside
my door.
Fuzzy Central Limit Theorem:
Data whose variation is due to many small independent random
influences will have an approximately normal distribution.
Balls and pins, heights of women, etc.
Questions on HW: Probabilities involving sample means.
This took the whole class. Will start
here Wednesday.
* * * * * * * * * * * * * * * * * * * * * * * *
* * * * *
Chapter 6, Introduction to Inference
Statistical Inference: drawing conclusions about a population from
sample data.
Requires: Random sample or Randomized
experiment. (Simple random sample usually)
First example: Use sample mean xbar
to "estimate" (unknown) population
mean µ
Mean of 4 grades (HW#4.40) estimates population
mean of all 10 ("known"= 69.4) E.g. 69.75, 64.25,
73.5
-
"xbar IS µ" Never true exactly
-
"xbar is close to µ" True for most xbars, depending on "close",
and sample size n.
-
"xbar is probably close to µ" ["probably"?? It is or it isn't.
We have "confidence" ours is a close one],
"xbars are usually close to µ" True.
-
If we have only one sample -- one xbar, we have NO IDEA if ours is one
of the "close" ones.
-
Quantify "usually" and "close."
tradeoff between "close"(accuracy) and "usually" (confidence)
Interval estimate: xbar + margin of error (fudge
factor) estimates population mean µ (69.4)
69.75 + 1: "µ is between
65.75 and 73.75" True
64.25 + 5:
"µ is between 59.25 and 69.25" False
Confidence interval estimate
of a(n unknown) population parameter:
-
an Interval constructed from the data, +
-
a Confidence level C:
C = probability that intervals constructed by this method will
capture the true, unknown, parameter.
Confidence Interval of the form estimate
+
margin-of-error for the mean with Confidence level C:
(p.306)
-
the estimate is xbar
-
margin of error m is : z* times Standard deviation
of sample mean
z* from normal table. Probability C is between -z*
and +z*. (Table A, or Table C, t dist. bottom row)
Standard deviation of sample mean: Sigma /sqrt(n)
Must know standard deviation of population!
or, If sample size is large, use s (standard deviation calculated
from sample)
The Birkenstock box contains numbers from a normally distributed population,
with population standard deviation 2.
You each constructed a 60% confidence interval for the unknown mean:
n = 4.
Standard deviation of sample mean = 2/sqrt(4) =
2/2 = 1
z* for C = 60% is .841, so margin of error m is
.841 times 1= .841.
How many people captured the true mean? (
previous
classes, 11/20 = 55% , 22/29= 76%.
9/18 = 50%.
Monday, 9/18 = 50%. Combined, 51/85=
60% It's unusual for it to come out exactly on the button.)
Why does the formula work?
-
If a particular xbar is within m of the population mean, then the
interval xbar + m contains the population mean.
& If a particular xbar is farther than m from the population
mean, then the interval xbar + m doesn't contain the
population mean.
-
We choose z* (and from it m) so that the probability
that Xbar is within m of the population mean is
C.
How? Probability C is between -z* and +z* in the standard normal
table,
between -z* ·(s.d. of Xbar) and +z* ·(s.d.
of Xbar) around µ, in the normal distribution of Xbar.
-
Table C, bottom row, is a restating of table A, normal table,
but with probabilities (areas) on the edge, and z values in the body.
To get the z* for C = 60% from the normal table, note that this
is the
middle 60%, which leaves 40% to be split between the 2 tails.
So 20% above z*, and 80% below. Go into the body of
table A, find .8000 is between values .7995 and .8023, closer to .7995.
The z value with .7995 below it is .84. Table C gives it more precisely
as .841.
Assumptions: pp. 312-13
SRS--other random samples get other formulas. Nonrandom
or biased samples can't use C.I.
Sometimes we can plausibly think of data
as SRS from large population (rolling dice, repeated weighings on scale)
Xbar is normal! OK IF 1) population is normal, or 2) n
big enough for Central Limit theorem.
Outliers? Trouble (xbar is sensitive).
Slight outliers ok (see next)
Skewness? n> 15 allows CLTh to overcome
all but strong skewness.
Sigma for population is known. Rarely true in practice.
Large n? substitute s calculated from sample. Small n--Ch.
7.
PreClass assignment Day 28 for Day 29
| Activstats: Same as given on Day 26 --work
on Confidence Interval stuff, either Activstats or Moore. |
HW Day28 Read (& reread)
Sec.
6.1
Memorize the tan box on p. 242 (mean and
s.d. of sampling dist. of x-bar)
the definition
of a C.I. p. 302, esp. the "repeated samples" bit, or above,
and the formula p.306 for a CI for the mean.
Closed Book Quiz Friday
or Monday on the two Confidence Interval things.
Note on reading: p. 306, table at top: "Tail area" is area
in One tail, which is what you look up in the Normal table.
Sec.6.1 , all from
Moore . These will all be assigned Wednesday, plus more, due Friday.
p.302, 6.1 poll of women
6.2 95% confidence?
6.3 density of x-bar, and confidence intervals
This problem combines the pictures 6.2 and 6.4 For part d, to draw
the confidence interval: just choose any point on the horizontal
axis of your graph to be x-bar. Measure off the distance m
(half the width of the shaded interval) and extend a bar m wide to the
left and the right of your point,below the curve. (Like fig. 6.4,
the bars with arrows at the ends. The red dots show what the x-bar
is for that confidence interval) Choose another point, and repeat..
If your first x-bar was in the shaded interval, pick your second outside
the shaded interval, and vice versa. You should note that if x-bar
is in the shaded interval, then the confidence interval bar covers mu (280)
and if x-bar isn't, then the bar doesn't.
-- - - - - - - - - - - - - - - - - - - - - -
- - - -
Using formula p. 306 for C.I.:
6.6 potassium again.
6.7 comparing CI's for different confidence
levels. Also write down the m (margin of error) for each interval.
6.9 comparing CI's for different sample sizes.
6.5 IQ test scores Read pp. 312-13 before
doing this one. |
Read, to discuss |
Optional |
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