Math 151 , Day 32, Wednesday, April 21, 2004Hit reload ...

Closed book quiz at end of class.   Conf. int. quiz
Definitions again:
HW questions?

Relation of m(margin of error, half width), C (confidence level), and n (sample size),  (and sigma)
    review Day31

Planning ahead before a survey:  Choose sample size big enough to satisfy desired: margin of error, confidence level.
Given C and m (and sigma), find n.
   Method:  Use C to find z*.  Plug in to formula for m, and solve for n.  Or memorize formula for n and plug in to it.
,    n = (z* sigma / m)²
     Note:  z* sigma still on top.  m and n change places, and whole thing is squared!
Example:  Suppose we have a normal  variable whose standard deviation  is 1.3 and we want to find a 90% confidence interval for it with a margin of error less than .2

Using table C we find that z* for a 90% confidence interval is 1.645.  Therefore
        n =  [(1.645)(1.3)/.2]2   =  114.3

        so we can use n = 115

           Round up!  If you get n = 5.06, you need a sample of size 6 to get your margin of error at least as short as you want. Finding sample size:  Memorize formula for n, or solve for n in formula for m.

"Statistics means never having to say you're certain."
Confidence interval Estimation made our best guess at an unknown population mean.
Testing will investigate a claim made that the unknown mean is actually a particular value.
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Sec 6.2: "Significance tests use an elaborate vocabulary, but the basic idea is simple: an outcome that would "rarely" happen if a claim were true--is good evidence that the claim is NOT true." (p.314)
(New terms?)

Shoeboxes (white and yellow slips): Take a sample of size 4 from each,  record, return numbers.
   I claim the mean value  for both shoeboxes is µ = 20.  Am I telling you the truth?  I can't remember for sure.  I do know that the distribution in the box is normal,  standard deviation is 4.
I do remember that if  µ is not 20, then it is greater than 20. µ > 20.
Take a sample of size 4, find xbar.   Once for each shoebox! (should have found xbar already)
How far from 20 is it?  far enough that I believe the mean is not 20??

Measure your xbar's distance from 20  in standard deviations of Xbar's. (That is, find z for xbar, assuming µ = 20. Note s.d. for sampling dist of xbar is 2 (why?) ).
Is this a far-out value of z? Look in the normal table to see how much probability is in the tail to the right of it--gives a measure of far-out-ness independent of distribution.

The game:
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.  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.
    Stated usually as: The parameter  is >, or <, (one-tail tests) --
                       or NOT = the particular value. (two-tail)
    H_a:   µ  > 1000 hrs. (Suppose we have a New process that makes them burn longer. We hope.)
    Other possible alternatives: H_a:   µ  < 1000 hrs.  (Want evidence that Mfr.'s claim is inflated)
             (two-sided=two-tail) H_a:   µ  Not = 1000 hrs.  (Want evidence that Assembly line process is"off")

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.
    The smaller the P-value, the stronger the data's evidence against H₀ ( for H_a).

For a test of µ  , using xbar (sigma known), the P-value is
--the area of the tail beyond the observed xbar, in the direction of H_a (one tail)
(--or twice that area (two-tail).)
We usually calculate it by standardizing the observed xbar (assuming H₀ true) and looking in the normal table. (p. 329)
How far from 20 is your xbar? Find z for xbar.
Is this a far-out value of z? What is the probability of being farther out, i.e. being in the tail beyond this z?  That's the P-value.

Start with understanding "null and alternative hypothesis, p-value."   Those are the foundation. Then

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)"
A particular scientific discipline may have a commonly accepted set of benchmarks, and language to go with it. (I think I remember .05 = "significant", .01 = "highly significant" in psychology?)
We will be less doctrinaire, use the language "significant at the alpha = ___ level." 
(However, "nobody" uses a significance level less rare  than .10, 1 in 10).