MATH 251, Probability and Statistics I, Fall 2001, Fri. Nov. 9, Day 30

--Questions on Confidence Intervals.
Sample size?  Round up: if the answer is n=23.2, you need 24 to achieve at least the desired m and C.
General pattern:  estimate + z* sigma_estimate
--Questions on upcoming quiz?

Significance tests 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)

Shoeboxes 1 and 2:  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!
How far from 20 is it?  Measure that in standard deviations of Xbar. (That is, find z for xbar.  Note s.d. for sampling dist of xbar is 2 (why?) ).  Is this a far-out value of z?

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.)

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 mu, 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).