| Moore
Ch. 14, Day 35 Hand in Sample
size for C.I. (& review of CI computations)You can check using the bottom section of the Confidence
Interval Excel sheet. A. New Shoeboxes: On a
Separate sheet: (2 shoeboxes. )The shoeboxes are outside my door if you missed doing them
in class. For each sample of size 4 from a
shoebox, write down the values, find the mean, (know which box you got them from: White #s,
green box. Yellow, red top.) and tell whether you believe
the population mean for that box is 20, or something bigger. (Your gut
feeling.) Does it help to know that the standard deviations for the
shoeboxes are both 4? Bring your sample numbers and xbars to class to pool, and Keep for further computations.)
(This is related to Chapter 15, where we'll learn the formal methods.) |
Read,
to discuss |
Optional A few problems good to review for the exam (repeated) p. 419, 17.7 Day care, parameter or statistic p. 422, 17.27 and 28 means vs. individuals. In #27, they're taking the "about what range" to be the interval containing the middle 99.7%--almost all. (Answer to last question of #28 is "no"--histogram of individual values in sample will be distributed (roughly) like the population.) p. 421, 17.26 WAIS, n = 1, n = 60 (Answers: a) about .3707, b) 100, 1.936, c).0049, d) a could be quite different; b still correct, c approx. right bcs of Central Lim. Th.)) |
Buffer
against
one low hour exam:
The final % exam grade minus 10 points will be substituted for the
lowest hour exam grade, if it is higher.
| Examples: | Ex1 | Ex2 | Ex3 |
Ex4 | final % | final -10 | |
| Student 1 | Original | 85 | 80 | 85 |
60 | 85 | 75, replaces lower 60 |
| Treated | 85 | 80 | 85 |
75 | 85 | <-- These will be used. | |
| Student 2 | Original | 85 | 80 | 80 |
70 | 75 | 65, lower than 70, don't replace. |
| Treated | 85 | 80 | 80 |
70 | 75 | ||
| Student 3 | Original | 85 | 50 | 75 |
55 | 85 | 75, replaces lower 50 |
| Treated | 85 | 75 | 75 |
55 | 85 | <--These will be used |
(Table
A, or Table C,
t dist. bottom row)
In practice: pp.
388-391
SRS--other random samples get other formulas.
Nonrandom
or biased samples simply can't do C.I.
Sometimes we can plausibly think of
data
as SRS from large population (rolling dice, repeated weighings on
scale)
--
For experiments, randomizing into groups allows us to use the methods;
but be careful about generalizing far beyond our "volunteers" type.
Ask how reasonably "like" a SRS the sample is.
Xbars are 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? "Moderate" sample size allows CLTh
to
overcome
all but strong skewness. (Numbers for "moderate" in Ch. 18)
Sigma for population is known. Rarely true in
practice.
Large n? Could
substitute s calculated from sample as "good" estimate of
sigma.
Small
n--Ch. 18, a slight modification of these methods takes care of unknown
sigma.
,
n = (z* sigma / m)2
Why does the CI formula work? (optional) Not done Mon.
"Statistics means
never having to say you're
certain."
Confidence interval Estimation made our best guess at an
unknown population mean.
Testing will investigate a claim made that the
unknown
mean is actually a particular value.
~~~~~~~~~~~~~~~~
Ch. 15: "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.363 top)
Suppose someone claims that the average height of Wells women over the
years is 70" (5'10"). I take samples (151 classes) every
year. Last term my sample has mean 65.67" (n = 20ish). Standard
deviation for heights of women in population is supposed to be about
2.5" , so s.d. for means from samples of 20 is about 2.5/4.48=
0.56.
IF the real mean is 70", my sample is astonishingly unusual:
z =(65.67-70)/0.56= -4.33 /0.56 = -7.73, 7.73 s.d's below the
mean. Conclude the claim is Not true.
Did this Mon.
- - - - - - - - - - - - - - - - - - - - -
- - -
Extended Standard Normal Table--"Normal Tails"
(also from Weblinks
page, )
z
P(Z <
z)
P(Z > z) = same in scientific notation: E-03 = 10-3
3.00
.9986501019683700
.0013498980316301 1.35E-03
4.00
.9999683287581670
.0000316712418331 3.17E-05
5.00
.9999997133484280
.0000002866515718 2.87E-07
6.00
.9999999990134120
.0000000009865877 9.87E-10
7.00
.9999999999987200
.0000000000012799 1.28E-12
8.00
.9999999999999990
.0000000000000007 6.66E-16 Below this, machine
can't compute.
If your assumptions lead you to a(n almost)
impossible
z value, question your assumptions!
(The basis of significance/hypothesis testing)
| Sievers home | Math151-Sp08/Days34.htm | 2pm | 4/21/08 |