MATH 251, Probability and Statistics I, Fall 2001, Wed. Nov. 28, Day 36 final version

7.1 continued.pp. 518-523. Sign test and log transformation, when the data's clearly not normal.  (remaining cell in table)
Log transformation can sometimes turn right-skewed data normal. (A distribution for which this is true is said to have a "lognormal" distribution.)  The only unsatisfactory thing is that you can't translate your confidence intervals back to the original units. Mean of log(x_i's) doesn't equal log(mean of x_i's).

Will cover Sign Test Friday:
Sign test is a nice "trick", that turns any paired sample situation into a binomial situation. For each pair, "success" is that the item from group 1 is bigger.  If there are ties, just throw them away (like the flipped coin that balances on its edge).  The null hypothesis is always that the groups are the same, so it is just like a coin-flip, the prob. of success is 1/2 under H₀. Then see how likely you are to get at least as many successes as you saw, using the binomial distribution.  That's the p-value.
     Disadvantage:  You're obviously throwing away a lot of information (how big the differences are).  The result is that the power to detect a difference--if there is one--is much less than that of a t-test, where the t is usable.
     The sign test can be extended to a single data set, where you test the median:  If a is the median, then in the population, half the observations will be above a, and half below. Each data point is then like a coin flip, above or below the median.  (Can you see how this could be extended to test for a particular value of the first quartile, for instance?)
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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. 242.
                /--- 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,
We need the Standard Deviation  of  xbar₁ - xbar₂ , and then we can proceed as before, more or less.
Using the Algebra of means and variances, we find that
      Variance (xbar₁ - xbar₂) = Variance (xbar₁) + Variance (xbar₂)
So theStandard Deviation is calculated like the hypotenuse of a right triangle (Pythagorean Theorem),  from the individual standard deviations.  We can use this to standardize the difference (xbar₁ - xbar₂), and get a  standard normal Z (p. 539).
But usually the standard deviations are unknown, and we substitute s's for sigmas.  Then our hypotenuse formula is

    SE_diff  = sqrt(SE(xbar₁)² + SE(xbar₂)² )

"t" = (xbar₁ - xbar₂)-(µ₁ - µ₂)       (See p. 541, for another way of writing the same thing.)
              SE_diff
   It would be nice if substituting s's gave us a t distribution.  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.550.  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 in practice.  However, the pooling of the data to estimate the common variance is a device also used elsewhere, so is worth looking at.
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Read pp. 518-523.
Read ahead 7.2, pp. 537-549, then continue.  We'll do all of 7.2.  You will NOT need to remember the d.f. formula p. 549.