Math 151 , Day 30, Friday, Nov. 8, 2002Hit
reload to get current versionAfter
Class
HW Day30 Read (& reread)
Sec.
6.1, read ahead, Sec. 6.2 at least to p. 334. I will ask you
next time to tell me some new words we'll need to understand.
(I have a MISPRINT p. 329, tan box, first formula::
z = x -BAR minus mu-sub-zero (etc.), not z = x minus mu-sub-zero (etc.)
as written. If your book is newer, this
may be fixed.)
the definition
of a C.I. p. 302, esp. the "repeated samples" bit, or below,
and the formula p.306 for a CI for the mean, including the picture explaining
C<-->z*.
Closed Book Quiz Monday,
see below
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.
Separate sheet: For each
of
your sets of 4 shoebox #s, find the mean, and tell whether you believe
the mean for that box is 20, or something bigger. Bring to class
to pool. The shoeboxes are outside my door if you missed doing them
in class.
- - - - - - - - - - - - - - - - - - - - - - -
Cautions; general review and extension
p. 314 6.14 internet, response rate
p.317, 6.19 newts
p. 318, 6.22 men/women CI's
p.316, 6.18 consumers/pharmacies
- - - - - - - - - - - - - - - - -
Sample size for C.I., review
p. 3.11, 6.10, 6.11, 6.12
p. 315, 6.16 enlighten the unstatistical
6.17hotel mgrs. |
Read, to discuss
6.13, 6.15 (cautions)
6.23 (Carter election)
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Optional |
Notes re Activstats: bottom of this
page.
Additions to Birkenstock box results:
2 of 2 "60% CI's" captured µ
: total (66+2)/(107+2)
= 62.4%
New Shoeboxes: Take 4 from each, write
them down (White from green box, yellow from red-top box) For HW, find
means (for each box separately.) Know which box! Does
that box have pop. mean 20, or some number >20?
HW questions?
Closed
book quiz Monday: 1)
Give a definition of a level C Confidence Interval for a parameter.
2) a) Write down the formula for a level C confidence
interval for the unknown mean of a normal population.
(Assume
the standard deviation of the population is known.)
b) Tell or show with a picture
how "C" connects with your formula.
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 µ with Confidence level
C:
(p.306)
Formula:
-
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)
m = z* (sigma)/ sqrt(n)
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.
Relation of m(margin of error,
half width),
C (confidence level), and n (sample size),
(and sigma)
C and z* get bigger and smaller together
(bigger C means bigger z*, and vice versa) (standard normal sketch)
m = z* (sigma)/ sqrt(n)
Want bigger C? Must accept bigger
m. Trade off confidence vs. accuracy.
But bigger n will make smaller m. This
makes sense: bigger sample size, more info-->more accurate estimate.
(square root makes it Expensive: have to quadruple n to make m half as
big)
So smaller m can be achieved only by
» accepting lower
confidence level (smaller C),
» or by increasing
sample size (bigger n).
Visual example: Author website, whfreeman.com/scc,
Applet: Confidence Intervals.
You can change the C, with the same
xbars, see the m change.
Sigma: We can't change it, it comes
with the population. But bigger sigma (more population variability)
will give bigger m (wider CI), i.e. less accuracy in prediction
(for the same C and n).
Science projects directed by Prof. Wahl: Experiments
on chickens bred to be "identical"--very low variability from one to the
other. Therefore very small samples suffice.
Planning ahead: Choose sample size big enough to
satisfy desired: margin of error, confidence level.
Given C and m (and sigma), find n.
Method: Use C to find
z*. Plug in to formula for m,
and solve for n. Or memorize
formula for n and plug in to it.
n
= (z* sigma / m)2 Note: z* sigma
still on top. m and n change places, and whole thing is squared!
Round up! If you get n = 5.06, you need a sample of size 6 to get
your margin of error at least as short as you want.
~~~~~~~~~~~~~~~~
Start here Monday. Sec 6.2: "Significance
tests use an elaborate vocabulary, but the basic idea is simple:
an
outcome that would "rarely" happen if a claim were true--is good evidence
that the claim is NOT true." (p.314)
Activstats for Ch. 6: DO Ch 16 if
you didn't. P. 16-3 is the same as 16-2 with an added activity, +
some new comments.
Ch. 17 (optional) introduces Confidence
intervals. (Alternative, read Moore 6.1) It can give you some
good reinforcement for what we are doing.
But many instructors (me included) think his demos with ybar in the
center of the
distribution may give a misleading sense of the
definition of confidence interval (tho his words are right.). So
if you work with
this chapter, be sure to do p. 17-4 (Appendix)
after p. 17-1.
Activstats Ch. 19--Hypothesis tests. It's
good, but not in exactly the order we're doing it.Optional
Notes for p. 19-1: Act uses "hypothesis test"
for what Moore calls "significance test". This is common.
activity 1: Try it again. Each time
it will reset the bar proportion to a different value.
activity 2: Reasoning of hypothesis test.
I think it's good, except it uses "unlikely" in the sense of "not very
believable" when it's talking about P(Red) = .50. , and we are trying to
use "likely" only about random phenomena. It's not a random phenomenon
whether P(Red) = .50. In any particular experiment, it either is,
or isn't, and if I peek inside the computer program I can tell you which--no
probability about it.
activity 3: Here the use of "unlikely" is
correct--we're taking a sample from the population, so our proportion is
a random phenomenon.
activity 4: A succinct exposition of the reasoning
and terminology.
p. 19-2 ACT does only 2-sided tests. But
it's good.
activity 1 is video--therapeutic touch
activity 2 is data on above. p=.4something
activity 3 gives a chance to organize the
info and do it
activity 4 introduces alpha level..
activity 5 relates C.I. and hypoth test but I
think it's confusing: "C.I.contains those values for which we would
not reject Ho." Skip it if you like.
p. 19-3 Goes through actually doing tests, p-value
and alpha.
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