Math 151 , Day 41, Wednesday, May 7, 2008  .After class, corrections..  Hit reload .

HW Day41:  Finish Reading Ch. 18: First to p. 441, then the rest.  READ about Matched pairs (p 444-7)and Robustness (p.447-9)! Check (new) p. 451, 23, 24. 
Read the Front page of the SPSS handout, compare the output shown there with the results in the book with the same example.  Learn to read the output!

Review Ch.9, p. 219 and around (Completely randomized experiment, especially with 2 treatments only), and p. 224 (Matched pairs experimental design) .
Read Ch. 19, pp. 460-61 only!  (Comparing 3 or more independent groups requires Analysis of Variance, Ch. 25)

Homework questions?  Day 40

Ch. 18:  Inference for population mean (realistic)
What we did: See Day 40 for more detail (optional) 

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, SE_Xbar, 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 Z 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.

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:     where t* is a little larger than the corresponding z*.
(By hand, we get t* from n-1 row of Table C, instead of z* from bottom row. )

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", except we need a table for "t" instead of Z (Table C)

New today:
Mostly with real data, you can let computer packages do these computations. Excel t-procedures
Get  SPSS Handout for Ch. 18 ( white folder outside my door)
   Look at Front page output, decipher it. Note Std.ErrorMean (standard error of the mean), t.
What SPSS calls "Sig. (2-tailed)" = "2-sided P-value"
If you have a one sided alternative, and your xbar is in the correct direction, divide the SPSS Sig. by 2 to get P.

MATCHED PAIRS t procedures-- "Paired samples"(SPSS), "Paired comparisons" Review: Ch.9 p. 224
   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.
The cola loss-of-sweetness example (SPSS handout, example 18.3, p. 440) was actually matched pairs: each "loss" number was a before-after difference; they just didn't tell us the before numbers or the after numbers.
    By hand, See Day 40, bottom

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. 447-449.   Assumption:  Population is Normal.
(Didn't quite finish talking about this Wed. Read the details yourself)
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. )

If you can't do t-procedures, there are procedures involving medians, or other approaches (Ch. 26)

 Another situation which uses t-statistics is the one in Chapter 19
"Two-sample problems".  Two random samples,  independent of each other, from distinct  populations. (Populations are normally distributed)
Often--comparing means from an experiment with two treatments (usually  "control" and "treatment"). Review Ch.9, p. 219 and around.
                 /--- Group 1, n₁---- Treatment 1---\
               /                                    \
 Random asst.(?)                                       Compare results --"means"
               \                                    /
                \--- Group 2, n₂---- Treatment 2---/
To examine  the difference of the  two means, µ₁ - µ₂, we look at the difference of the xbars. 
We need the Standard Error of the difference  xbar₁ - xbar₂ , and then we can proceed as before, more or less (with some adjustments.)
But enough already....