Math 151 , Day 33, Friday, Nov. 15, 2002Hit reload to get current version after class

Quiz:  If you got below 8 you can repeat it (same quiz) for a max of 8 points, today after class or by appointment.
HW Day33  ReRead (finish) 6.2 (pp337-8 optional), and read 6.3, especially pp. 343-345 for today.  Skip or skim Sec. 6.4.
       (Ch. 7 next.)
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 Monday from Moore  More p-values  p.341, 6.44 CEO pay.  Keep a copy of your z test statistic for use in 6.48 below. = = = = = = = = =  You should be able to do the * ones tonight:  But hand all the following  in Wednesday. Table C:  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 = = = = = = = = = =  *p. 342, 6.52 1% vs 5% *  6.53 define stat. signif. *p. 343, 6.54  knife edge .05 *p. 345, 6.55 and 56 effect of n

Read,  to discuss

Optional

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 are 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 for all distributions and parameters):
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. (Historically older language than P-value)
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)"

Start here Monday
Results of shoebox tests
HW questions:  #6.35, p. 333 Engine crankshafts:
Meaning of "significance"  (note--"High" significance means small alpha or P-value.)
Question: How do we know that .05 is "significant?" (.05 is 1 in 20 chance of seeing the result by "dumb luck" if the null hypothesis is true.)  Read sec. 6.3, pp. 343-345
>>Significance levels vary by field of study; different fields have different "customarily acceptable" levels.
      In reality, no sharp border between "significance" and "not significant"
>>How small a P is "convincing evidence" against H₀?  In practice...
        How plausible is H₀?  H_a?  Strong evidence needed to reject "conventional wisdom."
        How expensive (mentally, economically) will abandoning H₀ be?
>>"Statistically Significant" doesn't always mean "Important."  Big enough sample sizes will allow you to distinguish even small differences.
- - - - - - - - -
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 gives 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!)

= = = = = = = = = = = = = = = = = = = = = =
"Significance testing" vs. "Hypothesis testing"--gathering evidence vs. making decisions.