Math 151 , Day 34, Monday, April 26 2004Hit reload ... After class

If you see "statistically significant" without a level, it often means "at the .05 level".
I suggest this for each problem that you find a P-value or sig. level for: Sketch the curve representing the sampling distribution of x-bar, or of the z you calculate from x-bar, and mark your observational result on it (like fig. 6.10, 6.11, 6.13)

Hand in Wednesday  *can be done without table C. *p. 342, 6.52 1% vs 5% *  6.53 define stat. signif. + + + + + + + + + + + + (Sec. 6.3 ) *6.83 Train Welfare mothers This kind of study was the basis (plus conservative philosophy) for our present "welfare reform." *p.348 6.59 what is significance good for? * 6.60  radar detectors - - - - - - - - - - - - - - - - - - *Review of ch. 6--these  review CI and test (hand in Wed.):  p. 339 6.40 job satisfaction, 2 sided p. 360 6.74 wine--stemplot, CI , test.  Notice "less sensitive" noses will have higher thresholds. p. 362, 6.79 a,b effect of sample size = = = = = = = = = =  Next:  Table C: (Try these: Due Monday but may help with understanding) p.341, 6.48 CEO pay again (what you would do if you didn't have Table A) p. 341, *6.46, 6.49 general z statistic, significance,Turn the page--6.49 continues.  p. 342 *6.50 patent protection; another z. = = = = = = = = = =  Fixed significance levels: if you only have table C, what can you say?  p. 337, 6.37 testing number generator 6.38 nicotine content

Read,  to discuss  *p.346, 6.57  call-in poll *p.348 6.61 strong vs. signif.

Optional

Quiz back: Part b mostly good: 0 is the middle of the standard normal distribution.
Part a ranged from very good to too vague to just wrong things.  Definition

>EXAM 3  this Friday, Day 36, April 25, closed book.
Ch. 4 +Ch. 6,  through today's HW
(Through 6.3, p. 346 (not multiple analyses, etc.).  Table C NOT required, pp. 334-7 although covered today.  Not "tests from CI's", pp. 337-8.  )
Sample exam Handed out Friday.
     All Sig. test problems can be done without table C, tho problem 2 suggests it.
Solutions  outside my door (3), on reserve (2).

Summary so far:
Significance testing:   "an outcome that would "rarely" happen if a claim were true--is good evidence that the claim is NOT true."

Before taking data, define
H₀:"Null hypothesis" A claim or statement about the population we would like to show is NOT true.
       Stated usually as:  A parameter = a particular value.
H_a: "Alternative hypothesis" A claim or statement about the population we're trying to find evidence FOR.
          Stated usually as: The parameter  is >, or <, (one-tail tests) --
              or NOT = the particular value. (two-tail)

Take data.  Calculate statistic (outcome).  Is it an unlikely result if  H₀ is true?  Then that is evidence against H₀.
Measuring the strength of the evidence against H₀ (a common measuring stick) :
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. 321.
    One sided test:  size of Tail further out than observed value.
    Two-sided test: you need to measure the P-value symmetrically both directions from the observed value
           --so the P value is double what it would be for a one-sided test.
    The smaller the P-value, the stronger the data's evidence against H₀ ( for H_a).

A "Significance level" alpha is a probability level we decide on  in advance as being the "rarely" amount that will push us over into believing (well, sort of) that the H₀ claim  is not true.
We tend to use simple benchmark numbers for it, like .10 (1 in 10), .05 (1 in 20), .01 (1 in 100).
When the P-value is less  than (or equal to) a particular significance level alpha (say .05), we say,
    "The results are significant at the alpha = .05 level," or "The results are significant (P< .05)"

    >In reality, no sharp border between "significance" and "not significant." (P = .0499, P = .0511)
    >"Statistically Significant" doesn't always mean "Important."

Results of shoebox tests:  From dotplot after Friday's class:
    White #s (green box) H_o is true: 1/10 = 10%   of xbars found are sig. at 10%
                     (pooled with last year's, 3/33= 9.9% are sig. at 10%)
    Yellow #s (red top box) H_o is false:  8/11 = 73% are sig. at 10%
                     (pooled with last year's, 25/33 = 75.8%  are sig. at 10%
         If µ is bigger than 20 by a goodly amount, the test successfully detects this.
HW questions:  #6.35, p. 333 two-sided, + evidence for H_o  Engine crankshafts:
Note:  A Test requires a "random sample" (or --experiment-- random assignment of subjects.).  Otherwise Xbar is not a random variable, biases intrude, computations are worthless.  Read pp. 345-6.
- - - - - - - - - - - - - - - -
What if you don't have the Z-table but only have the t-table (Table C)?
What if you have a demanded level of significance, alpha?
    Table C: a limited list of probabilities  across the top row:
            = Right tail values for the bell curve distribution.
        The value in the bottom (z*) row under p is the corresponding standard normal value.
         "z* is the upper p critical value of the standard normal distribution."
  Do this: Find your z from the data. Make a sketch of the normal curve and mark z on it.  Mark the direction(s) of H_a.
    (If your z is in the direction of H_a , continue.  Otherwise the results are hopelessly not significant: you can quit.)
Find the two z*'s in Table C that bracket your z (ignore minus sign).  Find the corresponding p's.
    e.g. z =2.111
p      .02     .01
z*   2.054 \/ 2.326
       z = 2.111

So the P-value for your z is: between those 2 p's (one sided test)
                                           between double those 2 p's (two sided test)
    Test is significant at the bigger bracketing probability; not sig. at the smaller.
One sided: P-value is less than .02 and greater than .01
        Significant at the .02 level,not at the .01 level
Two sided: P-value is less than .04 and greater than .02
        Significant at the .04 level,not at the .02 level
If you have a specific demanded significance level, compare it with these levels.
            If  a test is significant at level b, then it is significant at every level bigger than b.
            If a test is Not significant at level d, then it is Not significant at every level smaller than d.
    "Significant at a":  probability of getting my results (again) by chance (if H₀ is true) is less than (or =) a.
       Significant at    Not significant at
p bigger  .10      .05      .01      .005     .001 smaller
                         /\
                        P-value
                        z-value (one-sided)
z* smaller 1.282   1.645  | 2.326    2.576    3.091 bigger
  You can compare z directly to z* for your desired alpha. The 2-sided is a bit tricky.
          (2-sided: Split the alpha in 2, then find the z*.  Don't halve or double z's--it doesn't work!)
Got to here.
= = = = = = = = = = = = = = = = = = = = = =
"Significance testing" vs. "Hypothesis testing"--gathering evidence vs. making decisions.