| Hand in: (SPSS problems may be handed in Wednesday,
if desired. I encourage you to try them now...)
Two-sample-- (SPSS problems are marked. Others by hand.) p. 396, 7.30, 7.31 s, SE, d.f. A. (SPSS)
(Mimicking
the handout.) Examples 7.7, 7.8, 7.10 in Moore, p.393 ff. Produce
(& Hand IN) the output shown in the handout. Write down the p-value
for the test, & the 90% confidence interval for the difference of means.
We'll
"always" use the "equal variances not assumed" option.
p. 401, 7.34 beetles in oats (test)
(SPSS) p.399 7.32 logging If you type in the data , remember you need all the tree species numbers in one column, and a "groups" column for logged or unlogged. (You can use strings or numbers for your logged/unlogged labels) 7.35 (a) chicks. You can do this efficiently by hand with a back-to-back stemplot, or use SPSS. SPSS won't do back-to-back stemplots, but you can get separate stemplots and side-by-side boxplots, using Analyze>Descriptive Statistics>Explore, using the plots there. Your response variable goes in the Dependent list, your groups variable goes in the Factor list. (SPSS) 7.35 b,c chicks Reading other output:
p. 400, 7.33 Math sublimina.(SPSS)
This is a complicated design: matched pairs, then 2-sample on the
differences! But notice chicks (7.35 ) was also matchedpairs--weight
gain =after-before--but they gave us the pre-subtracted numbers.
|
Final exam Final
exam is scheduled for Tuesday, Dec.17, 9-12 Please contact me ASAP
if you have a problem/conflict.
Exam is closed book and notes, except
bring One sheet of notes (both sides if you want) with anything you
want on it.
Get handout of info, and review problems if you
haven't.
~~~~~~~~~~~~~~~
Questions on HW:
Matched Pairs: Differences are often more
normal in shape than the separate variables ("weirdness" is often the
same for both items in a pair, and disappears in subtraction. Another
reason why this is a nice experimental design.)
Sec. 7.2, Comparing two means
"Two-sample tests". Two SRS's, independent, from
distinct
populations. (Populations are normally distributed)
Often--comparing means from an experiment with two treatments (usually
control and "treatment"). Cf. p. 140.
/--- Group 1, n1---- Treatment 1---\
/
\
Random asst.
Compare results
\
/
\--- Group 2, n2---- Treatment 2---/
To examine the difference of the two means, µ1
- µ2:
We need fairly normal populations; no extreme outliers.
Back to back stemplots are good; boxplots will do.
We use the difference of the two x-bars, (xbar1 - xbar2)
= diff.
We need the Standard Error of xbar1 - xbar2
,
and
then we can proceed as before, more or less.
The Standard Error is calculated like the hypotenuse of a right triangle
(Pythagorean Theorem), from the individual standard errors.
SEdiff = sqrt[SE(xbar1)2
+ SE(xbar2)2 ] P. 394 has another way of
writing the same thing.
"t" = (xbar1 - xbar2)-0
SEdiff
Unfortunately, this doesn't quite have an exact t-distribution, and
its exact distribution is very hard to deal with.
For doing by hand: df
= smaller of (n1- 1) and (n2- 1).
Will give a "conservative" result--slightly wider C.I., slightly less
significance, than a "sharper" value. If your results
hinge on the difference between this result and the computer result, they're
too close for comfort anyway.
From a computer: df = complicated formula on p. 403. Produces non-integer degrees of freedom. Very good approximation to the exact distribution, if both sample sizes are at least 5. Unsuitable for doing by hand.
Once we have (xbar1 - xbar2) , SEdiff
, and the df, our formulas pattern on the earlier ones.
Example
CI : estimate + t* . SEestimate
CI for µ1 - µ2,
difference
of means, is (xbar1 - xbar2)
+
t* . SEdiff
Test: H0: µ1 - µ2
= 0 same as µ1 = µ2 , "no difference"
Ha:
µ1 - µ2 > 0 same as µ1
>
µ2 Be careful with these, that you
know which direction you want.
or Ha: µ1
- µ2 < 0 same as µ1 < µ2
Often
we label our variables "1" and "2" so that we expect µ1
>
µ2
or Ha: µ1
- µ2 <> 0 same as µ1 <>
µ2 (not equal)
Calculate t, find P-value
(approximate, conservative)
--SPSS will do our computations when we
are given raw data.
Handout for SPSS two-sample, section 7.2
(backside is optional: tables built in
to SPSS)
We use the Not-equal-variances line of the results.
Monday I'll discuss this briefly:
There is a third way of doing these; the "pooled
two-sample t-procedure ." =="Equal variances assumed" (See Moore
p.406.) It was the only choice in many circumstances before the above good
approximations were developed, computing power increased, and robustness
was explored. The newer ways are usually preferable.
| Activstats Optional: two-sample
with pooled variance, Activstats ch. 21-3. Don't neglect the green
star.
This was the only technique until "recently." It's the basis for the much used Analysis of Variance, for comparing more than 2 independent groups. |
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