Math 151 , Day 35, Wednesday, April 25, 2001  final ver.

What is the significance to Statistics of the Can that is passing around the classroom?

Questions on Chapters 4 and  6
~~~~~~~~~~~~~~~
Ch. 7, Inference about population mean mu (sigma unknown).
"One-sample" procedures:  SRS of size n.  Use Xbar to estimate mu.
If we substitute s for sigma in the standardizing formula, we get t instead of z, with n-1 degrees of freedom.
Questions on HW, ch 7--using t-table.   Demo?
 
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.

Confidence intervals:  xbar + t* (s/sqrt(n))   Choose t* from table C, using the n-1 row, and confidence level C.
    Special case of common pattern:      estimate + t* SE_estimate

Significance tests:  State hypotheses as in Ch. 6, find t from data.  Use table C to find P-values for the t's it's between, write "P-value is between ___ and___".
(Or use software which will find P-value exactly. )

Example:  : bacteria per milliliter in10 specimens of milk raw milk from one producer.  Parameter: actual mean bacteria/ml.
         5370, 4890, 5100, 4500, 5260, 5150, 4900, 4760, 4700, 4870
4|5               n = 10,   xbar = 4950,  s = 268.45   SEM = 268.45/sqrt(10) =268.45/3.162=84.89   deg. of freedom = 9
4|77                    90% CI:  from t(9) in table,  t* = 1.833   CI is 4950 + 1.833^x268.45/sqrt(10)
4|889                                                                                      4950 + 1.833^x84.89, or  4950 + 155.6
5|11
5|23                   Test:  H₀ : mu = 4800                                       t = (4950 - 4800)/SEM = 150/84.89 = 1.767
                                    H_a : mu > 4800  (too contaminated)               t is between 1.383 and 1.833   (d.f. = 9)
                                                                                                         P is between .10 and .05.  Moderate evidence for H_a

MATCHED PAIRS t procedures:
   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 the null hypothesis will be "mu = 0", there is "no difference" between the treatments.
Example:  wax paper sandwich bags:  Is the wax layer the same inside and out?
25 bags:  measure (wax outside - wax inside) for each.  (pounds per square foot).
    Differences:   xbar = .093,  s = .723      SEM = .723/5 = .1446
    H₀ : mu = 0 (mean difference is 0)                  t = (.093 - 0)/SEM = .093/.1446 = .643.
    H_a : mu Not = 0 (there is a difference)            t is less than .685 (d.f. = 24) which is right-tail t* for probability .25
                                                                         Because test is 2-sided, double the tail: .50.  P value is > .50.
                                                  No evidence for difference.

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. 379-80.
t-procedures are quite robust against nonnormality.  But sensitive to outliers. Look at data.  Need SRS
 Details:  n<15   t ok unless data clearly not normal, or if there are outliers.
              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).

Wednesday  SPSS.  Bring your SPSS book. Check the webpage, Day 37 or 8, before class to see if we meet in Mac 101 or in 321.
HW Day 35, for Wednesday.  Read 7.1 for class Monday final version

Hand-in, due Wednesday  Sample size -->t*  p. 373, 7.4  CI,  7.5, 7.6 test, one- & two-sided  7.7 DDT  Find the mean and standard deviation by hand!(only 4 points) and do the rest by hand.       Make a note of your results; we will do this on SPSS too, check the results.  p. 386, 7.19 Shrimp ATP  A common calculational mistake is to divide the SE by square-root-of-n.  But square-root-of-n is already IN SE!  Don't divide by it again!  (I.e. pay attention to the difference between "standard deviation" and "standard error")  Matched pairs :  you just treat the difference/change as one variable (x). p. 378 7.8 tomatoes.  Give two values between which P lies, from Table C.  p.386  7.21 healing in newts You would only  need SPSS for part a, to check the mean and s.d.-- just look at the answers in the back of the book for them. Finish a, do b,c .  Robustness, etc. (text pp. 379-381)  --Make a dot plot of the differences in problem 7.21 p. 386 7.20 Acculturation  7.23 Increase in CEO pay  7.25 Presidents' ages

Read, to discuss

Optional  Review: