| Hand in on your separate
sheet:
A. For each of your samples of size 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. --Find 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) --Is your P-value smaller (less likely) than alpha = .10? If so, 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. + + + + + + + + + + + + + + + + + + + + + Start now, Hand in Friday :Sketching xbars for H0, p-value p. 323, 6.25 SSHA 6.26 Spending on housing - - - - - - - - - - - - - - Stating null and alternative hypotheses p. 325 6.27, 28, 29, 30 - - - - - - - - - - - - Calculating p-value (one-sided), relating to Sig. level p. 328, 6.31 and 32 (extending 6.25 and 26) 6.33 restating jargon |
Read,
to discuss |
Optional
(more practice) + + + + + + + +
|
Relation of m(margin of error,
half width),
C (confidence level), and n (sample size),
(and sigma)
review Day-30
Finding sample size: Memorize formula for n, or solve for n in
formula for m.
~~~~~~~~~~~~~~~~
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)
(New
terms?)
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??
Measure your xbar's distance from
20 in standard deviations of Xbar's. (That is, find z for
xbar, assuming µ = 20.
Note s.d. for sampling dist of xbar is 2 (why?) ).
Is this a far-out value of z?
Look
in the normal table to see how much probability is in the tail
to
the right of it--gives a measure of far-out-ness independent of distribution.
The game:
Before taking data, define
H0: "Null hypothesis" A claim or statement
about
the population we would like to show is NOT true.
Stated usually as: A parameter
= a particular value. H0: µ =1000
hrs. (Average lightbulb life.)
Ha: "Alternative hypothesis" A claim or statement
about
the population we are trying to find evidence FOR.
Stated usually
as: The parameter is >, or <, (one-tail tests) --or NOT = the
particular value. (two-tail)
Ha: µ > 1000 hrs. (Suppose we
have a New process that makes them burn longer. We hope.)
Take data. Calculate statistic (outcome). 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. 321.
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. 329)
How far from 20 is your xbar? Find
z for xbar.
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.
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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