| Hand in Finish finding the results for your shoebox numbers, if you haven't! (see below for how) A. Simulation of shoeboxes: On a separate page, please! Use Applet: P-value of a test of significance How to. H0: µ =20. Ha: µ > 20 n = 4, sigma = 4 (do Update. Not Reset-- Reset returns to mean 0) The picture will show the sampling distribution of the mean (the Xbars) when the real mean is 20. a) First simulate the shoebox where the mean is actually 20. At the bottom of the picture, enter 20 in "the truth about the mean is" box. Do Generate Sample 10 times; each time record from the picture the xbar and the P-value for that sample. Make a dotplot of your xbars! Find the number, and the proportion, of your 10 in which the P value is < .10. (Example: if 2 of your 10 had P's at .10 or below, 2/10 = 20% would have the P value < .10. b) Now simulate the shoebox where the mean is actually 24. Leaving the top numbers the same, at the bottom of the picture, enter 24 in "the truth about the mean is" box. Do Generate Sample 10 times; each time record from the picture the xbar and the P-value for that sample. (If the xbar is too big to show in the picture just write "big" for xbar, and record the P-value for that sample) Notice the picture is still for the null hypothesis, but the sample values are usually "high". Make a dotplot of your xbars! Find the number, and the proportion, of your 10 in which the P value is < .10. Hand in your page, and Be ready to add your numbers and proportions to the circulating sheet on Wednesday. p. 366, 15.3 Anemia p. 366, 15.4 Student attitudes (15.2, was done in class, Day 35) 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, mostly) 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: P-value of a test of significance How? How to. Do the rest on a separate sheet and keep it: More Setups and Calculations. Use the Applet: P-value of a test of significance to check your work. Use Table A (normal table) to find P-value: For 18, 19, 37, 38, 42, 43,44, be sure to: Write down your H's. Make a rough sketch of the normal dist. when H0 is true and the direction(s) of evidence for Ha . Write down your xbar, your z. Mark your xbar and z on the sketch, shade the area which is your P. Find your P. Leave a line or two of space for using table C with your z's, which we'll learn how to do next time. p. 376, 15.18 Water quality p. 376 15.19 SAT Check the mean you calculate in the back of the book. p. 382, 15.37 IQ test scores (The mean from the data is 105.84) p. 383, 15.38 & 41 hotel managers p. 383 15.40 Sample size affects p-value Use the Applet: P-value of a test of significance to do both n = 18 and n = 75. Hit "Update" between. Use the same xbar of 17 in both (below the picture). Hit Show P. Notice the scale changes so the xbar value appears to move "out". Repeat, comparing n = 18 , n = 28, n = 38. Here the written scale on the x-axis doesn't change (so xbar stays in the same place) but the normal curve visibly narrows as n increases. p. 383 15.42 Supreme Court p. 383, 15.43 wrong alternative p. 383, 15.44 the wrong p p. 383, 15.45 Placebo effect, make hypotheses. |
Read, to discuss |
Optional (more practice)
|
Exam 4's not finished--sorry!.
Final exam: Tues. May 13, 7-10 pm (evening!)
If this is
a
problem for you, please email me soon.
Alternative--Tueday morning/afternoon?? Wed.
morning/afternoon?? (I'll pick one!)
Full exam schedule is at http://www.wells.edu/pdfs/finals.pdf
Registrar's page with link to this and other
good stuff: http://www.wells.edu/academic/regist.htm
"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)
Need machinery to analyze less "obvious" results--build in
effect
of
standard deviation and sample size
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 what we 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'll
go over two-tail P-values in more detail Wed.
We usually calculate it by standardizing the observed xbar (assuming
H0 true) and looking in the normal table. (p. 369 on)
<>Applet:
P-value of a test of significance automates this. (Uses "raw"
scale of xbars, rather than z-scores). How
to. Use as check, guide.
Continue with new material Wednesday!
Start with understanding "null and alternative hypothesis,
p-value." Those are the foundation. Then
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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