| Hand in Wednesday .
Bring also questions about Exam. Postpone again! Won't be on the exam. & & & & & Leftover problems from Day 30 & & & & & & & & These ideas are related to those in Ch. 15. p. 290, 11.39 Pollutants in auto exhausts For 11.39: You might want to know L so that if you tested your 25 cars and found a high value of x-bar, you would be able to compare it with L; if it was greater than L, you would go back to the manufacturer and say "I believe you sold me a batch of bad cars, because the chances of getting an average emission level this high if the exhaust system is working properly is only 1 in 100. It is more reasonable to believe the exhaust system is not working, than that we "are" that 1 in 100 possibility." p. 290, 11.38 Glucose testing If we use this cutoff level L to say that people (with a mean of 4 tests) over L "have diabetes", then the chances of declaring that someone "has diabetes" when they really are OK (with mean 125mg/dl) is .05. .05 or 5% is the chance of a "false positive" using this protocol, when the real mean is 125. & & & & & & & & & & & & & & & & & & <>Beginning Ch. 15 Do 15.1 p. 364, 15.1 Anemia Exam 4 material ends at this point. Do this one on a separate sheet: We haven't learned the words yet, but the directions don't assume you know the words. 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. *I'll leave the boxes outside my door, so If you didn't get your samples in class for any reason, you can come and get them. + + Postpone the rest.+ + + + + + + + + + + + + + + + + + + Stating null and alternative hypotheses p. 366, 15.3 Anemia p. 366, 15.4 Student attitudes (15.2 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.
Exam 4, Friday Nov. 17, Day 36. This Friday. As usual: One
sheet of notes; I will give you
tables. Covers Ch. 10 p. 257 on (Continuous models and R.V.'s),
Ch. 11 up to p. 286 only, Ch. 14, Ch. 16 to p. 391, Ch. 15 to p.
364. Sample
exam handed out last time;
solutions outside my door/on reserve today.
Jenn O'Neill writes:
my TA hours are Monday 6-8pm, Tuesday 3:30-5:30pm, Wednesday 6:30-8pm and
in addition this week I will be holding extra hours on Thursday 6:15-8pm for anyone who wants to review for the exam on Friday.
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 | <--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.
HW questions? Day
33
Many questions, reviewing
computations and concepts.
Postponed again: Look back at
11.36 and 38, p. 297. 38: "backward
normal" problem. From a proportion/probability, find a z*,
from that a raw value (here an x-bar).
Note that table C gives us another way to get z*'s for some
probabilities!. Bottom row, "one sided P". The table is set
up to go from "tail" probability to z*, without having to calculate
"probability to the left."
"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)
Start here next.-
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- -
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 . This is
3.6/6 = 0.6 s.d.'s above the mean, a pretty typical kind of value.
xbar =
125.8 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 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 >
1000 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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