Math 151 , Day 30, Friday, April 12, 2002Hit
reload to get current versionAfter
Class
Additions to Birkenstock box results: 2
"60% CI's" captured, 2 didn't: total (51+2)/(85+2) = 61%
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 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.
We'll start here Monday
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).
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 colloquium last Wednesday: 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.
~~~~~~~~~~~~~~~~
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)
PreClass assignment Day 30 for Day 31
Activstats: Same as given on Day 29--work
on Confidence Interval stuff, then
Significance Tests, either Activstats
or Moore. Moore--Read 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.) |
HW Day30 Read (& reread)
Sec.
6.1, read ahead, Sec. 6.2
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, including the picture explaining
C<-->z*.
Closed Book Quiz Monday,
see top of page
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.
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
- - - - - - - - - - - - - - - - -
With Monday's HW
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)
- - - - - - - - - - - - - - |
Optional |
This page belongs to Sally Sievers who is solely responsible
for its content. Please see our statement
of responsibility.