| Hand in Wednesday . Sample size for C.I. (& review of CI computations)You can check using the bottom section of the Confidence Interval Excel sheet. p. 356, 14.10 Estimating mean IQ p. 358 14.24 Hotel managers p. 360, 14.33 calibrating a scale Continue with shoebox numbers: on a separate sheet: If you haven't already, get a sample of size 4 from each of the two shoeboxes (in class, or outside my door.) (White from red-top box, Yellow from green box.): Bring Wed: A. For each of your samples of size n=4 from the two shoeboxes *(keep track of which box they came from!): test H0: µ=20 vs. Ha: µ > 20. Do it like this: --Find xbar (may have already). --Standardize your xbar, thus finding a z (assuming the population mean is 20, and the population s.d. is 4, so the s.d. for Xbar is 2) --Use the standard normal table to find the probability to the right of your z. (this is the "P-value" for your x-bar.) --Is your P-value smaller (less likely) than alpha = .10? (Y/N) If Yes, your result is "significant at the alpha = .10 level" --Do you think the box has mean > 20? Be ready to add your results to the circulating sheets Wednesday. *Boxes outside my door, if you didn't get your samples in class or over the weekend. <>Beginning Ch. 15 p. 364, 15.1 Anemia Stating null and alternative hypotheses p. 366, 15.3 Anemia p. 366, 15.4 Student attitudes (15.2, and more, done in class) p. 367, 15.6 travel time p. 367, 15.7 stating hypotheses - - - - - - - - - - - - Test statistic: xbar to z p. 368, 15.8, 15.9, 15.10 (same old examples) - - - - - - - - - - - - Calculating p-value (one-sided) p. 371, 15.12, 15.13, 15.14 (Same examples). Calculate by hand. p. 371, 15.11, Applet. Do the one given (two-sided), then check your answers for 15.12, 13, 14 (one-sided) using the applet. |
Read, to discuss |
Optional (more practice) + + + + + + + + |
Your shoebox
results: Write your xbars (one on each pad--yellow or
white) and make a dot for each on the circulating dotplot.
Exams not finished.
Final exam: Wed. May 16, 2-5 pm. If this is a
problem for you, please let me know soon.
Full exam schedule is at http://www.wells.edu/academic/dates.htm#exams
| 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 | <--ß 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 | <--ßThese will be used |
This is to encourage those who are nervous about Exam 4, and to
encourage all to try to put it
together for the final.
RECAP:
Confidence Interval of the form estimate
+
margin-of-error for the mean µ with
Confidence level
C:
(p.349-50)
(Table
A, or Table C, t dist. bottom row)
,
n = (z* sigma / m)2
Why does the CI formula work? (optional)
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.
"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. This year my sample has mean 62.81" (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
(62.81-70)/0.56= -7.19 /0.56 = -12.8, 12.88 s.d's below the mean.
Conclude the
claim is Not true.
- - - - - - - - - - - - - - - - - - - - -
- - -
Extended Standard Normal Table
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)
-
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- -
Need machinery to analyze less "obvious" results--build in effect
of
standard deviation (if s.d. were 10" would my sample still be
inconsistent with the claim?) and sample size (if n were only 4 would
that change my result?) .
Do 15.2 p. 365: Normal, s.d. = 30. Claim:
pop. mean = 115. n = 25. IF mean is really
115, Xbars are N(115, 6). Sketch!
xbar = 118.6 .
118.6-115=3.6. This is 3.6/6 = 0.6 s.d.'s above the mean, a
pretty typical kind of value.
xbar =
125.8 125.8-115=10.8. This is 10.8/6 = 1.8
s.d's
above the mean, high enough to be pretty unusual if the mean is
really 115.
xbar = 139
139-115=24. This
is 24/6 = 4 s.d.'s above the mean, unreasonably high if the
mean is really 115.
So 125.8 or (more so!) 139 would be evidence that the mean
for this group (older students) is NOT 115, is in fact higher.
Shoeboxes (white and
yellow
slips): Take a sample of size 4 from each,
record,
return numbers.
I claim the
mean value for both shoeboxes is µ = 20.
Am I telling you the truth? I can't remember for sure. I do
know that the distribution in the box is normal, standard
deviation
is 4.
I do remember that if µ
is not 20, then it is greater than 20. µ > 20.
Take a sample of size 4, find
xbar. Once for each shoebox! (should have found xbar
already)
How far from 20 is it?
far enough that I believe the mean is not 20??
Take data. Calculate test statistic,
usually based on one that estimates the parameter in the
hypotheses. For µ, test statistic is the z-score of xbar,
so a big z-score number means that xbar is far from µ.
Is it an unlikely
result if H0 is true? Then that is
evidence
against
H0.
Measuring the strength of the evidence against H0 (a
common measuring stick for all distributions and parameters):
P-value of
a test: The probability, computed assuming
that H0 is true, that the observed outcome would
take a value as extreme or more extreme than that actually observed
(if
we could repeat taking-data again). p. 368.
The smaller the P-value, the stronger the data's
evidence against H0 ( for Ha).
For a test of µ , using xbar (sigma
known),
the P-value is
--the area of the tail beyond the observed xbar, in the
direction of Ha (one tail)
(--or twice that area (two-tail).)
We usually calculate it by standardizing the observed xbar (assuming
H0 true) and looking in the normal table. (p. 369 on)
H0: µ =20 Ha:
µ > 20 How far from 20 is your xbar?
Find
z for xbar.
For xbar = 24, z = 2
Is this a far-out value of z?
What
is the probability of being farther out, i.e. being in the tail beyond
this z? That's the P-value. P
= .0228
A "Significance level" alpha is a probability level
we
decide on in advance as being the "rarely" amount that
will
push us over into believing (well, sort of) that the H0
claim is not true. (Historically older
language
than P-value)
We tend to use simple benchmark numbers for it, like .10 (1 in 10),
.05 (1 in 20), .01 (1 in 100).
When the P-value is less than (or equal to) a particular
significance
level alpha (say .05), we say,
"The results are significant at the alpha = .05
level," or "The results are significant (P< .05)"
A particular scientific discipline may have a commonly accepted set
of benchmarks, and language to go with it. (I think I
remember
.05 = "significant", .01 = "highly significant" in psychology?)
We will be less doctrinaire, use the language "significant at the alpha
= ___ level."
(However, "nobody" uses a significance level less rare
than .10, 1 in 10).
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