Math 151 , Day 39, Friday, May 4, 2001  proposed

"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."  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.
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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 uses it to produce the "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 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 problems are marked (SPSS); do all others by hand.  Remember: For a 2-sample (independent samples) procedure, SPSS (and all other statistical packages I know) require all the response data (the stuff to be compared--breaking strength of polyester) in one variable, with another variable telling which group each case belongs in (how long buried).

Hand-in Monday : Get handout on Two-sample t procedures ...two independent samples (SPSS)  p. 391, 7.28, 7.29 which design? p. 396, 7.30, 7.31 s, SE, d.f. A.  (SPSS) (Mimicking the handout.) Go thru Example 7.5, p. 102-104 (buried polyester), in the SPSS manual, matching up with Examples 7.8,9, 10 in BPS, p.396 ff.  Produce (& Hand IN) the output shown in the SPSS manual table 7.5, and write down the p-value for the test, & the confidence interval for the difference of means.   We'll "always" use the "equal variances not assumed" option. Note--If you bring in the data from the Excel or text file, the "groups" column will have 2's and 16's, not "2W", "16W", so you use 2 and 16 as the labels for group 1 and 2.  If you use 2 for group 1 and 16 for group 2, it will do (2-week-mean) - (16-week-mean).  It will also allow you to use 16 for group 1 and 2 for group 2--then it will do (16-week-mean) - (2-week-mean).  So (unlike the matched-pairs situation) you can choose which way to subtract. p. 401, 7.34 beetles in oats (test) p. 412, 7.49 voice onset time (test and CI) (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.404, 7.37 (DDT),  p.406, 7.39 self concept = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =  More complicated problems: putting together everything... Will be assigned Monday or Wednesday p. 400, 7.33 Math sublimina. 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. p.410, 7.45 fitness  Do b.  Then Look in the back at the answers for a and b. p.422, 7.63 pasture fertilization p.423 7.67 London bus people p.425 7.72 reading biology