MATH 251, Probability and Statistics I, Fall 2005, Fri. Nov. 4, Day 30After Class, corrected

Exam 2: Open book takehome.  Available probably Friday Nov. 11 (Day 33).  Due Before I leave for Thanksgiving break (Day 37)

Quiz Monday: Knowing and using: Binomial distribution formula (p. 349 bottom)
mean and st. dev. for Binomial:  X (count), p-hat (proportion)   (Summary p.350)
mean and st. dev. for X-bar from SRS of size n   (Summary p. 368)
Normal?  Central limit theorem: says yes for all of the above distributions, approximately, for n large.
If population(s) normal to start with, linear combinations stay normal (including X-bar), mean and s.d. follow algebra rules (as last quiz.)
Questions on quiz?

Homework questions? Day 29
Continuing 6.1
Common general pattern for CI:   estimate + z* sigma_estimate (e.g. for proportion p:  p-hat + z* sigma_p-hat )
Level C confidence interval for the mean, with margin of error m,
where P(-z* < Z < z*) = C
Day 29: why does it work?

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 (standard normal sketch)
Want bigger C?  Must accept bigger m.  Trade off confidence (Big C)  vs. precision (small m).
    OR: 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 halve m)
    So smaller m can be achieved only by   » accepting lower confidence level (smaller C),
        » or by increasing sample size (bigger n)               » or by decreasing sigma.

Visual example:  Applet,  Confidence Intervals.You can change the C, with the same xbars, see the m change.  (n and sigma are fixed.) (50 intervals display. More will overwrite the drawings of old ones.  But they accumulate numerically  in the right panel till you reset.)

    Sigma:  We can't usually 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).  Ways to affect sigma: 1) measuring--be very careful. Better instrument?  2) Science  projects directed by Prof. Wahl:  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.
     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.
   (What about sigma?  Estimate from a pilot study or similar situation.)
Start here Monday
Significance tests, Sec 6.2
Significance tests use an elaborate vocabulary, but the basic idea is simple: a result that would "rarely" happen if a claim were true--is good evidence that the claim is NOT true."
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Extended Standard Normal Table
  z         P(Z < z)                         P(Z > z)   = same in scientific notation: E-03 = 10^-3
3.00   .9986501019683700    .0013498980316301    1.35E-03
4.00   .9999683287581670    .0000316712418331    3.17E-05
5.00   .9999997133484280    .0000002866515718    2.87E-07
6.00   .9999999990134120    .0000000009865877    9.87E-10
7.00   .9999999999987200    .0000000000012799    1.28E-12
8.00   .9999999999999990    .0000000000000007    6.66E-16
Below this, machine can't compute. If your assumptions lead you to an (almost) impossible z value, question your assumptions!
(The basis of significance/hypothesis testing)
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Shoeboxes 1 and 2:  I claim the mean value  for each shoebox 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?) ).  Did you get 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.)
     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 a Test statistic (result).  Is it an unlikely result if  H₀ is true?  Then that is evidence against H₀ (and for H_a.)

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 test statistic would take a value as extreme or more extreme than that actually observed (if we could repeat taking-data again).  p. 405.
    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. (pp. 409-12)

Example:  H₀: µ =1000 hrs.  (Average lightbulb life.)  Design a competing bulb:   Show it's better.
                 H_a:   µ  > 1000 hrs.
        Sample of size n = 25.  Population sigma = 150 hrs. Get xbar = 1060 hrs.  Are these bulbs better?
            Draw a picture of  the distribution of X-bars, assuming H₀  is true;  shade the region of P-value P(Xbar>1060)
                 z = (1060-1000) ÷ (150/5) = 2.
         P(Z > 2) =  .0228  More than  2% and less than 3% chance of getting a result this high if we did it again.
   Different question:  quality control of "old" assembly line.   H_a:   µ  Not = 1000 hrs.
      Same picture, but P-value is symmetrical tails, P(Xbar < -1060) + P(Xbar>1060).  As "far out" in either  direction.
         P-value = 2P(Z>2) = .0456.  More than 4% and less than 5% chance of getting a result this far out if we did it again.

Shoebox:  H₀: µ =20       H_a:   µ  > 20    (if it's not 20, I know it's greater.)
How far from 20 is your xbar? Find z for your xbar:  (xbar - 20)/ 2, since s.d. of xbar is 2, and µ =20 under the null H₀.
Is this a far-out value of z? What is the probability of being farther out, i.e. being in the right hand  tail beyond this z?  That's the P-value.

Null and Alternative hypotheses:    Both are statements about the population.  But Not a symmetrical situation.
    Want evidence FOR the Alternative.  "New is better than old (maybe just different); change has taken place; a relationship does exist," etc.
     The Null is a "straw man" to be knocked down; the position of the skeptic who has to won over.  "New is no better/the same as the old.  No change has taken place.  No relationship exists."
Null Also needs to be something that, if true, gives a specific distribution of the test statistic; so we can tell if the test statistic is "unusual" compared to what's expected under the Null hypothesis.
So Null is Usually a specific parameter value that the test statistic is an estimator for.
             H₀: µ =1000 hrs;   test statistic Xbar estimates µ; so Xbar should be close to 1000 if H₀  is true.

In doing research, much effort should (& does) go into creating appropriate and testable hypotheses, and clarifying the data and test statistic that will be used.

Next:  "Statistical significance."