Math 151 , Day 30, Friday, April 13, 2001

Monday's class starts here:        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) (hand in lists of new terms)

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:  mu =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:   mu > 1000 hrs. (New process makes them burn longer. We hope.)

Take data.  Calculate statistic (outcome).
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)

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).

Continue to Reread Sec. 6.2.  Focus on pp. 318-334 for next time.  When that's under control, continue.
Sec. 6.2:  Work on these (at least read them to know the issues); Hand in Wednesday

Hand in:  Sketching x-bars for H0, p-value  p. 323, 6.25 SSHA  6.26 Spending on housing Stating null and alternative hypotheses  p. 325 6.27, 28, 29, 30  Calculating p-value (one-sided), relating to Sig. level  p. 328, 6.31 and 32 (extending 6.25 and 26)  6.33 restating jargon  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.

Read, to discuss

Optional  (more practice)  Stating null and alternative hypotheses  p.340, 6.41,42  Calculating p-value (one or two-sided), using z test statistic, relating to Sig. level  p. 340 6.43 watered milk?