Math 151 , Day 31, Monday, April 15, 2002Hit reload to get current versionAfter class

Closed book quiz at end of class.   Day30
HW questions?

Relation of m(margin of error, half width), C (confidence level), and n (sample size),  (and sigma)
    C and z* get bigger and smaller together (bigger C means bigger z*, and vice versa) (standard normal sketch)
m = z* (sigma)/ sqrt(n)
    Want bigger C?  Must accept bigger m.  Trade off confidence vs. accuracy.
    But bigger n will make smaller m. This makes sense: bigger sample size, more info-->more accurate estimate.
            (square root makes it Expensive: have to quadruple n to make m half as  big)
    So smaller m can be achieved only by
        » accepting lower confidence level (smaller C),
        » or by increasing sample size (bigger n).

    Sigma:  We can't change it, it comes with the population.  But bigger sigma (more population variability) will give bigger m (wider CI), i.e. less accuracy in prediction (for the same C and n).
Science colloquium last Wednesday:  Experiments on chickens bred to be "identical"--very low variability from one to the other.  Therefore very small samples suffice.

Planning ahead:  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!
            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.
~~~~~~~~~~~~~~~~
Start here Wednesday
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 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?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.)

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).

Continue to (Re)read Sec. 6.2.  Focus on pp. 318-334 for next time.  When that's under control, continue.
Sec. 6.2:  Look these over to see where we're going.  Hand in Wednesday.

To hand in Wednesday: Moore Sketching xbars 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?