Hand in Wed . Finish finding the results for your shoebox
numbers, if you haven't! (Day 37, A) |
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.
Final exam: Thurs.
Dec. 13, 9-12am. If this is a problem for you, please email me soon.
Alternative--Tueday Dec. 11 morning/afternoon?
Full exam schedule is at http://www.wells.edu/academic/dates.htm#exams
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 37 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: 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.
+ + + + + + + + + + + + +
Try the homework on Significance Level; we'll
continue to talk about it Wed.
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 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!)
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