|
Hand in Fri . Rearranged: do the Setups and Calculations section, leaving space for the Table C results. Postpone the rest. If you didn't, do A from Day 37
(shoebox simulation) Postpone all "significance"
problems: "Significance"and table C: |
Read, to discuss Postpone: 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. Friday seems sure..
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!) Wed.
morning is current favorite.
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
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 35 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
37: 15.8, 9, 10
Measuring the strength of the evidence against H0 (a
common measuring stick for all distributions and parameters)--how weird
is my observation if H0 is true?:
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. Table A
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). Table A
HW questions? Day 37: 15.12, 13,
14 (one-sided). 11 (two-sided.)
Example (one sided again): H0:
µ =1000 hrs. (Average
lightbulb life.)
Suspect company's cheating:
Show mean is 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 far
below 1000 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:(two
sided) 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.
+ + + + + + + + + + + + +
Start here Friday
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 far out (in this
direction) 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:
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.
- - - - - - - - - - - - - - - -
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, prob. .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 standard 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, ever!--it doesn't
work!)
| Sievers home | Math151-Fall08/Days38.htm | 1pm | 5/7/08 |