Math 151 , Day 40, Friday, Nov. 30, 2007 .After class  Hit reload .

Your real shoebox results; z's, P's, sig at alpha's, to the circulating lists.
Your simulation of shoebox results:  (25 each, for mean of 20, mean of 24) 
To the circulating pad: Add your total # where P < .10, (your # of simulations: should be 25),
            and proportion (total # with P<.10)/25.

A little review of regression

Reviewof significance testing, in brief: 

Before taking data, define
H₀: "Null hypothesis" A claim about the population we would like to show is NOT true.  
   A parameter = a particular value.  H₀: µ =1000 hrs.  ("Average lightbulb life".)
H_a: "Alternative hypothesis" A claim or statement about the population we are trying to find evidence FOR.
   The parameter <, or > the particular value (one-sided/tailed)  Or NOT= the particular value (two tailed).

Take data.  Calculate test statistic. For µ, test statistic is the z-score of xbar. (Start with xbar, standardize using mean of H₀)
    Is it an unlikely result if  H₀ is true?  Then that is evidence against H₀.
Evidence: how strong?  P-value of a test:  The probability, computed assuming that H₀ is true, that the observed outcome would take a value as extreme or more extreme than that actually observed (if we could repeat taking-data again).  p. 368. 
Small P = strong evidence.
   One-tail alternative:  P = the tail beyond the observed value, in the direction of H_a
   Two-tail alternative:  P = the sum of both tails, farther out than the observed value in either direction.
Results are significant at level alpha  if P < alpha, not otherwise.  Significance levels are usually "benchmarks."   What's "statistically significant" can vary by field.  (.05 is usually good.)

Calculate for your shoebox numbers!

Cautions: see Day 39 for details.
Homework questions?  Day 39
   16.9, p. 395:  "Statistically insignificant"--the differences could easily be due just to chance variability.  Why is it important to know differences "small"?  Because a large difference could be "real" but "statistically insignificant" just because the sample size was too small to confirm it.
Other HW ??

Look at shoeboxes, and simulations so far. Dotplots of real shoeboxes.
For the real shoeboxes,  the white numbers (where the mean is really 20) rejected H₀ : µ = 20  (incorrectly) in favor of H_a : µ > 20
          at the alpha = .10 level in 3 of 18  samples (16.7%)(Fall '06)
                                               1of 16 samples(6.3%) (Sp. '07): 4 of 34 (11.8%) combined
                                               1 of 13 samples(7.7%) (Fall '07, 251):5 of 47(10.6%) combined
     the yellow numbers (where the mean is really bigger than 20--24 I think)   rejected H₀ : µ = 20 (correctly!)
          at the alpha = .10 level in 16 of 19  samples (84.2%) (Fall '06)
                                               13 of 17 samples(76.5%) (Sp. '07): 29 of 36 (80.6%) combined
                                               11 of 13 samples(84.6%) (Fall '07, 251): 40 of 49 (81.6%) combined

Got to here Friday. Will look briefly at Ch. 18 Monday.
Ch. 18:  Inference for population mean (realistic)
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 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.
    Table C: "critical" t-value in the body, probabilities at top and bottom.  Set up for P-->t.
       Example.  t(20) = 2.086  corresponds to 
                   Confidence level 95% = "middle" probability between -2.086 and +2.086
        one-sided P = .025,  probability in the one tail above +2.086 = probability in the one tail below -2.086
        Two-sided P = .05,  probability in the two tails beyond -2.086 and +2.086.
              (For z distribution, the corresponding z* is 1.96; notice t is further out.) 

     Excel t-procedures sheet will find P's from t's.

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:     Choose t* from table C,  n-1 d.f., level C.

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", only using the (n-1) d.f. row in  table C,
to find P-values for the t*'s it's between, write "P-value is between ___ and___".   (or Excel t-procedures)

Example: bacteria per milliliter in 10 specimens of  raw milk from one producer.
  Parameter: actual mean bacteria/ml.
       5370, 4890, 5100, 4500, 5260, 5150, 4900, 4760, 4700, 4870

4|5  4|77 4|889  5|11  5|23

n = 10,   xbar = 4950, s = 268.45   SEM = 268.45/sqrt(10) =268.45/3.162=84.89.  deg. of freedom = 9 90% CI:  from t(9) in table, t* = 1.833   CI is 4950+1.833x268.45/sqrt(10)                                                        4950  +1.833x84.89, or  4950+155.6 bacteria/ml. If we had KNOWN Population sigma = 268.45,    we'd have used z* = 1.645, gotten a narrower CI.   (but we don't know sigma!) Test:  H0 : µ = 4800                          t = (4950 - 4800)/SEM = 150/84.89 = 1.767           Ha : µ > 4800                           t is between 1.383 and 1.833   (d.f. = 9)              (too contaminated)                Table C: One-sided P is between .10 and .05.  Some evidence for Ha (If the test had been 2-sided, P would be between .20 and .10) Excel t-procedures:  P-value = .05552

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   n = 25    SEM = .723/5 = .1446
H₀ : µ = 0 (mean difference is 0)                  t = (.093 - 0)/SEM = .093/.1446 = .643.
H_a : µ 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 greater than .50.
                                           No evidence for difference.
       Excel t-procedures:   for t = .643, d.f.24, two-sided P = .526 
- - - - - - - - - - - - - - - - - - - - -
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. 447-450.   Assumption:  Population is Normal.
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. )