| Hand in MONday . A. Simulation of shoeboxes: Use Applet: P-value of a test of significance H0: µ =20. Ha: µ > 20 n = 4, sigma = 4 (do Update. Not Reset-- Reset returns to mean 0) 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 25 times; each time record from the picture the P-value for that sample. Find the number, and the proportion, of your 25 in which the P value is < .10. (Example: if 3 of your 25 had P's at .10 or below, 3/25 = .12 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 25 times; each time record from the picture the P-value for that sample. Notice the picture is still for the null hypothesis, but the sample values are usually "high". Find the number, and the proportion, of your 25 in which the P value is < .10. Hand in your lists of xbars, and Be ready to add your numbers and proportions to the circulating sheet on Monday. P-value and significance p. 372, 15.15, 16 (more of the same problems. #15, P = .1894., # 16, P = .0359 ) p. 372 15.17 ultramarathoners 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: then also see what P's your z is between according to table C. 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 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). Show P. Notice the scale change and the xbar value 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. &&Postpone yet again & & & 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. & & & & & & & & & & & & & & & & & & |
Read, to discuss For 15.35, p. 382: Ignoring the actual question: Which of the answers to 35 is self-contradictory? Which one makes logical sense (whether or not it's true)? Sketch a normal curve and mark out the areas for alpha = .10 and alpha = .05. |
Optional (more practice) |
Exams not finished.
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)
Day 34 for other details.
Summary, comments:
Take data. Calculate test statistic. For
µ, test statistic is the z-score of xbar. (Start with xbar,
standardize using mean of H0)
Is it an unlikely
result if H0 is true? Then that is
evidence
against
H0.
HW questions? Day
35: 15.8, 9, 10
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-sided)
(--or twice that area (two-sided).)
<>Applet: P-value
of a
test of significance automates this. (Uses "raw" scale of
xbars, rather than z-scores).
HW questions? Day
35: 15.12, 13, 14 (one-sided). 11 (two-sided.)
Example (one sided): H0:
µ =1000 hrs. (Average
lightbulb life.) Suspect
company's cheating: Show it's worse.
Ha:
µ < 1000 hrs.
Sample of size n = 25.
Population sigma = 150 hrs. Get xbar = 940 hrs. Are
these bulbs worse than claimed?
z
= (940-1000)
÷
(150/5) = -60/30 = -2.
P(Z
<
- 2) = .0228 = P-value
More than 2% and less than 3%
chance of getting a result this high if we did it again.
Example (two sided): H0:
µ =1000 hrs. (Average lightbulb life.)
Ha: µ Not = 1000
hrs. (Quality
control
on assembly line--find if it is "off" either way.)
Ha: "Alternative hypothesis"
A claim or statement about the population we are trying to
find
evidence FOR.
A value either much bigger than or much
smaller
than the H0 value is evidence against H0 &
for
Ha.
Sample of size
n = 25.
Population sigma = 150 hrs. Get xbar = 940 hrs. z
= (940-1000)
÷
(150/5) = - 2
P(Z
< - 2) = .0228
P-value: We measure the
probability
of seeing something (again) as extreme as the observed
value
(or more so).
So you need to measure the P-value symmetrically
both directions from the observed value--so the P value is double
what it would be for a one-sided test. P-value
is approximately 5%; more precisely, 2·.0228 =
.0456
So for a test of a mean, the P-value for one-sided is half
that for two sided, IF the result is in the direction of evidence for
the alternative.
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. Appropriate levels vary by discipline.)
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)"
. Giving actual P is better, if you can.
Lightbulbs: One-sided: .0228 = P-value. More than 2% and
less than 3%
chance of getting a result this high if we did it again.
"Significant at
the alpha =.03 level. Also at the alpha = .05
level" (P-value says, rarer than these levels)
"Not significant at the alpha = .02 level. Also not significant
at the alpha = .01 level" (P-value
says, more common than these levels)
Two-sided: .0456 = P-value.
(Barely) less than 5% chance of getting a result this far out if we did
it again.
"Significant at the alpha = .05 level. (Also at alpha =
.10). Not significant at the alpha = .04 level. Nor
.01 level.
Applet: Applet:
Statistical Significance
You can pick the alpha you desire, and see if your x-bar lies outside
the "alpha" barrier(s). (approach of p. 376-79) But P-value is more
informative.
HW questions? Day
35
- - - - - - -
Today, finally? NOT YET Look
back at
11.38, p. 297. "backward
normal" problem. From a proportion/probability, find a z*,
from that a raw value (here an x-bar). We can think of this as a
significance testing question. n = 4, sigma = 10 mg/dl.
H0:
µ =125mg/dl (Sheila is normal), Ha:
µ > 125 (Sheila has gestational diabetes.)
Find the L so that only .05 of
random samples of 4 tests would have mean above L, among
people(Sheila) whose real mean is 125.
L is the "cutoff" for doing an alpha =
.05 test. 5% of "healthy" people will be diagnosed diabetic
(false positive).
Doctors like a "decision
making rule", want an alpha cutoff to apply, rather than
calculating a P-value for each indivual's set of 4 tests..
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."
- - - - - - - - - - - - - - - -
YES
What if you don't
have the Z-table but only have the t-table (Table C)?
What if you have a demanded level of
significance,
alpha?
Table C: a
limited
list of probabilities across the bottom rows:
= Tail values for the bell curve distribution.
(one sided = one tail, two sided = two symmetrical tails)
The
value in the z* row above P is the corresponding
standard
normal value ("critical value").
Check z* = 1.960, .025 above it (or below -1.960). .05 farther
out than it. Corresponds to Table A.
Do this: Find your z from
the data. Make a sketch of the normal curve and mark your z on
it.
Mark
the direction(s) of Ha.
(If your z is in the
direction(s)
of Ha, continue. Otherwise the results are hopelessly
not significant: you can quit.)
Find the two z*'s in Table C that bracket your
z
(ignore minus sign). Find the
corresponding P's.
e.g. z =2.111
z = 2.111
z* 2.054 \/
2.326
One-sided
P ...
.02 .01
Two-sided P ... .04 .02
So the P-value for your z is: between .02 and .01
(If it's a one sided test)
&
between double those 2 p's--between .04 and .02 (If it's a two
sided test)
Test is significant at the
bigger bracketing probability; not sig. at the smaller.
One sided: P-value
is less than .02 and greater than .01
Significant at the .02 level,not
at the .01 level
Two sided: P-value
is less than .04 and greater than .02
Significant at the .04 level,not
at the .02 level
If you have a specific demanded
significance
level, compare it with these levels.
If a test is significant at level b, then it is
significant
at every level bigger than b.
If a test is Not significant at level d, then it is Not
significant
at every level smaller than d.
"Significant at a":
probability of getting my results (again) by chance (if H0
is
true) is less than (or =) a.
My result is less common than a.
Results
Significant at
Not significant at
p bigger
.10 .05
.01 .005 .001 smaller
/\
P-value
(one-sided)
z-value
z* smaller
1.282 1.645 |
2.326 2.576 3.091 bigger
You
can compare z directly to z* for your desired alpha. z
>z*? Significant at that alpha.
The 2-sided is a
bit
tricky. Don't
halve or double z's--it doesn't work!)
Have a lovely
Thanksgiving!
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