Math 151 , Day 34, Monday, Nov.15, 2004Hit reload ...

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 Calculating p-value (one or two-sided), using z test statistic, relating to Sig. level  p. 333, 6.34 price reduc. on coffee   6.35 crankshafts true? Use your calculator to find the sample mean.   6.36 cola? Use your calculator to find the sample mean. - - - - - - - - - - - - - - - - - - - - - -  More p-values  p.341, 6.44 CEO pay.  Keep a copy of your z test statistic for use in 6.48 next time. p. 343, 6.54  knife edge .05 p. 345, 6.55 and 56 effect of n  *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

>EXAM 3  this Friday, Day 36, Nov. 21, 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 probably covered today.  Not "tests from CI's", pp. 337-8.  )
Sample exam was Handed out Friday.  All Sig. test problems can be done without table C, tho problem 2 suggests it.
Solutions  outside my door (2), on reserve (2), should soon be on electronic reserve.

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.   (example soon)
    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."  (Do a C.I. to find out amount of difference)

Results of shoebox tests:  From dotplot after Friday's class:
    White #s (green box) H_o is true: 0/7 = 0%   of xbars found are sig. at 10%
                     (pooled with last 2 years, 3/40= 7.5% are sig. at 10%)
    Yellow #s (red top box) H_o is false:  6/7 = 86% are sig. at 10%
                     (pooled with last 2 years, 33/44 = 75%  are sig. at 10%
         If µ is bigger than 20 by a goodly amount, the test successfully detects this.
Homework questions?

Two-Sided example
Tonight's 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.
Results   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.