Math 151 , Day 36, Wednesday, November 28, 2001  final version

Meet in Mac 101 Computer Lab, with SPSS manuals, Monday!
Science Colloquium Friday:  Mathematics! Logical paradoxes, speaker from Cornell.
Questions on Exam material

Interesting news release:  Government "experimental methods" to measure tar and nicotine in cigarettes are not useful .  Smokers manage to smoke in such a way as to get more tar and nicotine, and low tar cigarettes have not helped with cancer rates.   http://newscenter.cancer.gov/pressreleases/lowtar.html
Also the National Cancer Institute's fantastic website of charts and graphs http://www.nci.nih.gov/atlasplus/charts.html
(lots of 95% confidence intervals)

Questions on 7.1, one-sample and matched-pairs t-procedures.

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.
    SE_diff  = sqrt(SE(xbar₁)² + SE(xbar₂)² )  p. 394, for another way of writing the same thing.

"t" = (xbar₁ - xbar₂)-0
             SE_diff
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) , SE_diff ,  and the df, our formulas pattern on the earlier ones. Example
CI :  estimate + t* ^. SE_estimate
    CI for µ₁ - µ₂, difference of means,  is (xbar₁ - xbar₂) + t* ^. SE_diff
Test:  H₀: µ₁ - µ₂ = 0 same as µ₁ = µ₂ , "no difference"
           H_a: µ₁ - µ₂ > 0 same as µ₁ > µ₂   Be careful with these, that you know which direction you want.
      or H_a: µ₁ - µ₂ < 0 same as µ₁ < µ₂ Often we label our variables "1" and "2" so that we expect µ₁ > µ₂
      or H_a: µ₁ - µ₂ <> 0 same as µ₁ <> µ₂  (not equal)
        Calculate t, find P-value (approximate, conservative)

There is a third way of doing these; the "pooled two-sample t-procedure."p. 406  It was the only choice in many circumstances before the above good  approximations were developed, computing power increased, and robustness was explored. It requires that the variances of the two populations be equal.  The newer ways are usually preferable.
~~~~~~~~~
HW  Read 7.2.  You are responsible for the material through p. 402; should read and understand the rest in order to be able to deal with the output from SPSS, and future encounters.
--You don't need to know the formula for d.f. on p. 403, only that SPSS (and other computer packages) use it to produce their "equal variances not assumed" result.  You would never try to calculate it by hand--too much possibility for a mistake..
--You don't need to know the "pooled two-sample t-procedure", only that it goes with the "equal variances" line in  the SPSS or other computer results--we prefer to use the "equal variances not assumed" results in all cases.  You should know that you will still meet the "pooled" procedure as the "standard" in  older books, or areas where the newer method has not filtered down yet.
--SPSS will do our computations when we are given raw data.  We'll learn how next week.

Hand in Monday : Meet in Lab, with SPSS Manuals Nothing to hand in Monday except the assignment from Monday Day 35 - - - - - - - - - - - -  The following will be assigned after we begin 7.2 p. 391, 7.28, 7.29 which design? p. 396, 7.30, 7.31 s, SE, d.f. p. 401, 7.34 beetles in oats (test) p. 412, 7.49 voice onset time (test and CI) Reading other computer output: p.404, 7.37 (DDT),  p.406, 7.39 self concept