Math 151 , Day 40, Monday, May 7, 2007 .After class.Data added 11am5/7 Hit
reload .
HW Day40: (Re)read Ch.
16, especially Multiple tests, pp 395-6
Read Ch. 17, p. 414 and p. 417 I
Reading Ch. 18: We'll repeat the CI and test work,
only with s instead of sigma, and t instead of z. First to p. 441,
Next, the rest. Read it all for Wednesday!. Check p. 451
18.15, 15, 17, 18, 19, 20, 21, 22
first, then 23, 24. Chapter 18 will be the last work of the
course.
Hand in Wednesday .
Review concepts
p. 423, 17.35 brains
p. 424, 17.36 support groups
p. 424, 17.37 CA brush fires
Review meaning of P, significance:
p. 384, 15.48 Cicadas
p. 385, 15.52 P?
p. 385, 15.53 sig. def.?
t- procedures
p. 434, 18.1 and 2,
s<-->standard error
p. 436, 18.3 Critical values: Use Table C and
also the Excel t-procedures sheet; be
sure your answers are consistent.
p. 436, 18. 4 Critical values: Use Table C
. For b, make a sketch. Note the decimal place is different
in (a) and (b)
p. 437 18.5 Critical values for CI.
p. 437 18.7 Ancient air CI Make a dotplot or
stemplot to examine the data. It will look somewhat skewed, but
with so little data this kind of scatteredness can happen easily from a
normal distribution. We should report that the skewness may make
our CI only approximately accurate. Xbar = 59.5889% and s =
6.2553% are what you would get if you calculated from the data; use
these to make your CI. Optional: Check with Excel t-procedures
p. 453 18.29 absenteeism CI
p. 455 18.36 a. Calcium and blood pressure CI (to
use
Table C where the degrees of freedom aren't given, go to the row with
the lower degrees of freedom, here 50. You're giving up a
little bit of sharpness rather than overstate your case.) Optional: To
see how much difference the "correct" t* would give, use Excel t-procedures)
p. 441 18.8 and 9 is it significant? Also, use Excel t-procedures to find the P-values
more exactly.
p. 432 18.25 read carefully. (one or more t-values was
incorrectly computed.)
p. 432 18.10 Ancient air test See note to
18.7, for mean and s.d. Optional: check with Excel t-procedures
= = = = = = = = = = = = = = = = = = = = = =
Postpone: Matched
pairs,
and robustness (by hand)
p. 455, 18.37 measuring placebo effect Use
Table C. You can check with the Excel
t-procedures
p. 450, 18.14 Reading scores Use Table C. Also, what IS the
standard
deviation? You can check with the Excel
t-procedures . Also , you may find that the mean is
(statistically) significantly below the basic level. Is the
difference large enough to be important? (I don't know...)
p. 455, 18.36b calcium/blood pressure conditions.
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Read,
to discuss
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Optional
(review)
p.424, 17.6 support groups
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Get Exam 4 back if
you didn't. Links to comments, solutions Day
39:
Final exam: Wed. May 16,
2-5 pm. Two sheets of notes: If this time is a
problem for you, please let me know soon.
You may start early (noon or later Wed.) Other arrangements,
please confirm with me!
Full exam schedule is at http://www.wells.edu/academic/dates.htm#exams
Review exercise (Open book, help from anyone. Optional)
will count for 50% of Final exam score. Due beginning of exam.
Available WED. I hope.
Late HW: accepted up to beginning of exam.
Your simulation
of
shoebox
results: (25 each, for mean of 20, mean of 24)
To the circulating pad: Add your total # where P < .10, (your # of
simulations: should be 25),
and
proportion (total # with P<.10)/25.
Look at shoeboxes, and simulations so
far.
For the shoeboxes, the white numbers
(where the mean is really 20) rejected H0 : µ
= 20 (incorrectly) in favor of Ha : µ
> 20
at the alpha = .10 level in 3 of 18 samples (16.7%)(Fall
'07)
1of 16 samples(6.3%) (Sp. '07): 4 of 34 (11.8%)
combined
the yellow numbers (where the mean is really
bigger than 20--24 I think) rejected H0 : µ
= 20 (correctly!)
at the alpha =
.10 level in 16 of 19 samples (84.2%)
(Fall '07)
13 of 17 samples(76.5%) (Sp. '07): 29 of 36
(80.6%) combined
<> Simulation: with mean = 20,
reject H0 : µ = 20 (incorrectly)
at alpha = .10 in
31/250 or 12.4% of samples (close to
.10) (Fall '07)
19 of 100
samples(19%) (Sp. '07): 50 of 350 (14.3%) combined
with mean = 24, reject H0
: µ = 20 (correctly!) at alpha = .10 in
181/250
or .724 of samples (Fall '07)
79 of 100 samples(79%) (Sp. '07): 260 of
350 (74.3%) combined
<> Shoebox
simulation, my set of 25 each. Your actual values on
overhead.
IF you use a particular alpha as a "cutoff" between "reject H0
" and "failing to reject H0"--we can talk about probability
of rejecting H0 when it's true--and alpha is that
probability! Applet:
"Power" (of a test to detect a difference).
Homework questions? Day
39
16.10, 40, 41: like the above. Also Day 39 notes
16.9, p. 395: "Statistically
insignificant"--the differences could easily be due just to chance
variability. Why is it important to know differences
"small"? Because a large difference could be "real" but
"statistically insignificant" just because the sample size was too
small to confirm it.
Other HW ??
Ch. 18: Inference for population mean
(realistic)
The most
unrealistic of our "simple conditions" for inference (p. 344) was that
we knew the population standard deviation sigma. We remove that
condition here.
If we substitute s, the sample standard
deviation, for sigma, the population standard deviation, in our
Normal distribution formulas:
If n is quite big, the value of the sample standard
deviation will be close to the same as the value from the
population, and our work's approximately right.
But if n is smaller, estimating sigma by s will add
in extra variability! Problem solved by modifying the
Z-distribution!
Standard error of the (sample) mean = 
Standard deviation of xbar, estimated from the data.
"Standard
error of the mean": s/sqrt(n) SEM, SEXbar,
etc.
When you estimate the standard
deviation of a statistic, the resulting
estimate is called the "standard error" of
the
statistic.
t-distribution
family: like standard normal only slightly fatter in the
tails, slightly more spread.
Mean = 0. Symmetrical around 0.
t(k)
is the t distribution
with k degrees of freedom.
Comparison with normal (Excel
graph)
Lower d.f.--fatter tails. Higher d.f.--more
like standard normal.
Table C: "critical"
t-value in the body,
probabilities at top and bottom. Set up for P-->t.
Example. t(20) = 2.086
corresponds to
Confidence level 95% = "middle" probability between -2.086 and +2.086
one-sided P
= .025, probability in the one tail above +2.086 = probability in
the one tail below -2.086
Two-sided P = .05, probability in the
two tails beyond -2.086 and +2.086.
(For z distribution, the corresponding z* is 1.96; notice t is further
out.)
Excel
t-procedures
sheet
will find P's from t's.
Standardizing xbar with s instead of sigma results in
the one-sample t statistic, t-distribution with n-1degrees
of freedom.
Conditions for inference about a mean: (p. 434)
++ SRS (or reasonable facsimile)
++ Population is Normal. (Can relax
to symmetric, single-peaked unless n "very small")
"One-sample"
t- procedures:
SRS
of size n. Use Xbar
to estimate µ.
Confidence intervals: 
Choose t*
from table C, n-1
d.f., level C.
Significance tests:
State hypotheses
as in Ch. 15, find
t from data, by:
Calculating the one-sample
t-statistic, using the null hypothesis value of µ (call
it
µ0)
Then
proceed
as if it were a "z", only using the (n-1)
d.f.
row in table C,
to find P-values for the t*'s it's between,
write
"P-value is between ___ and___".
(or Excel t-procedures)
Example: bacteria per milliliter in 10
specimens of raw milk from one producer.
Parameter: actual mean bacteria/ml.
5370, 4890,
5100, 4500, 5260, 5150, 4900, 4760, 4700, 4870
4|5
4|77
4|889
5|11
5|23 |
n =
10,
xbar = 4950,
s = 268.45 SEM
= 268.45/sqrt(10)
=268.45/3.162=84.89.
deg. of freedom = 9
90% CI: from t(9) in table, t*
= 1.833 CI is 4950+1.833x268.45/sqrt(10)
4950 +1.833x84.89,
or 4950+155.6
bacteria/ml.
If we had KNOWN Population sigma =
268.45,
we'd have used z* = 1.645, gotten a
narrower
CI. (but we don't know sigma!)
Test: H0 : µ
= 4800
t = (4950 - 4800)/SEM
= 150/84.89 =
1.767
Ha : µ >
4800
t is between 1.383 and 1.833 (d.f. = 9)
(too
contaminated)
Table C: One-sided P is between .10 and .05. Some evidence for Ha
(If the test had been 2-sided, P would be
between
.20 and .10)
Excel t-procedures: P-value
= .05552
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Next time,
optional: SPSS
for "raw data"
Next time:
MATCHED PAIRS t
procedures--
"Paired samples"(SPSS),
"Paired comparisons"
before--after, left
hand--right
hand, Drug A vs. Drug B on the same person or on a matched pair.
For each pair, find the difference
in the observed values. Then treat these differences as if
they
are "the" data set, from a normal population, and do One-sample t
procedures.
Usually (always?) the null hypothesis
will be "
µ = 0", there is "no difference"
between the treatments.
Example: wax paper sandwich
bags:
Is the wax layer the same inside and out?
25 bags: measure (wax outside - wax inside)
for each. (pounds per square foot).
Differences: xbar
= .093, s = .723 n =
25
SEM = .723/5 = .1446
H0 : µ
= 0 (mean
difference
is
0)
t = (.093 - 0)/SEM
= .093/.1446
= .643.
Ha : µ
Not = 0 (there is a
difference)
t is less than .685 (d.f. = 24)
which is right-tail t* for probability .25
Because test is 2-sided, double the tail: .50. P value is greater
than .50.
No evidence for difference.
Excel
t-procedures: for t = .643, d.f.24, two-sided P =
.526
- - - - - - - - - - - - - - - - - - - - -
ROBUST procedures: a confidence
interval or significance test is called robust if the
confidence
level or P-value doesn't change very much when the assumptions of the
procedure
are violated. pp. 447-450.
Assumption: Population is Normal.
t-procedures are quite robust against
nonnormality.
But
sensitive
to outliers, bad skewness. Look at data. Need SRS!!
Details: n <15
t ok if data roughly symmetric, single peak, no outliers. Don't
use if skewed or outliers.
(How out is an outlier?)
n > 15 t ok unless there is strong skewness, or
outliers.
n > 40 or so: t ok even if there is skewness.
(Outliers?
I suggest trying with and without them, see what changes).
Matched-pairs data (differences) are often more normal in
shape than the separate variables ("oddness" is often the same for both
items in a pair, and disappears in subtraction. Another reason
why
this is a nice experimental design.
)
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