MATH 251, Probability and Statistics I, Fall 2005, Fri. Nov. 11, Day 33After class

2) Assume µ =24, really!  (same setup as before, n = 4, sigma = 4 ).  If alpha = .10, Power = .761.  Change alpha up and down (smaller and larger than .10).  Notice that smaller alpha means smaller power to detect the alternative,  and vice versa--we're just moving the cutoff line in both pictures.  (Nothing to write down)

3) Assume µ =24, really!  (same setup as before). Set  alpha = .10. If n = 4, Power = .761, about a 3/4 chance of detecting that the mean is really >20.  Suitable for my class example, where I wanted some people to NOT detect it. Not so good for real work.
Find the power for n = 8, 12, 16, 20.  Graph n (4 to 20) on the horizontal axis and power on the vertical axis; connect the dots smoothly.  From your graph, what sample size should I use if I want 95% power (approximately)
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Exam 2: Open book takehome.  Available today (pink folder outside my door).
Due by 1 pm Monday  Nov. 21 (Day 37)  Covers Chapter 3 through 6.3. Start it now, since there's little HW tonight!

Quiz Monday: Wednesday! to write down these definitions (memorize them):  I want the general definitions, not just formulas for the tests of µ.
>>"A Level C confidence interval for a parameter is...._ __ __ __ __ __ __"
>>P-value for a test.
>>"A test is significant at level alpha = .05 if the P-value__________________"

Homework questions, 6.2? Day 32
Sec. 6.3, odds and ends about significance testing. Notes Day 32
Monday: finish Day 32 (6.3: multiple tests), then the brief look at 6.4 below, then 7.1.
Sec. 6.4

[New York Times, Nov. 13, 01--report on the finding of the first anthrax case in New York City:  The test that was first used was new; they hadn't had time to confirm the results by the usual method of growing a culture.]  Dr. Koplan, director of the Centers for Disease Control and Prevention, on the phone with Mayor Giuliani:      "Are you sure it's anthrax?" the mayor asked.      "Well, we have a high degree of probability," Dr. Koplan replied.      "No, no, no, don't give me that [stuff]," was the mayor's rejoinder. "Is it anthrax or is it not?'      "Yes," Dr. Koplan said.      "Fine, that's all I needed to hear," Mr. Giuliani said.

"Significance testing" vs. "Hypothesis testing"-- two different approaches that blur... (Sec.6.4)
Both start with null and alternative hypotheses.  You want to show the alternative is true.

Significance testing:  Calculate P-value (or closest alpha), describe how unusual your result is if H₀ is true.
Let the audience for your work decide if they believe in the alternative hypothesis or not.  (Scientist's approach.)
   Language: "strong evidence for H_a, against H₀"  or not strong...

Hypothesis testing:  Make a decision  between H₀ and H_a (often associated with predetermined fixed alpha level)
We need to do something.  (Business, government?)
    Language:  "Accept H_a, reject H₀" if P-value smaller than alpha.
        What if we can't reject H₀?  Do we accept H₀? Safer: "fail to reject H₀"
    H₀ "Innocent"                 "Guilty" H_a
                  \ "Not Proven" /         but defendant goes free...

If we make a decision we run the risk of error:
Type I error,  Accepting alternative H_a when null H₀ is true (probability = alpha)  Test designed to focus on this one.
Type II error, Accepting null H₀ when alternative  H_a is true (probability = beta, depends on what exact parameter value in  H_a is true)  Can't make this one if we refuse to commit, but
A small Type II error means the "power" of the test to detect  that the alternative hypothesis is true,  when it really is true-- is high.

		the	truth
		Ho is true	Ha is true (some particular value)
my	Reject Ho	Type I error (Prob. = alpha)	OK (Prob. = Power)
decision	Accept Ho	OK	Type II error (Prob. = beta)

                       Each column's probabilities are conditional on its "truth".  Probabilities in each column add to 1.
Power = 1 - P(type II error).  Varies with what actual parameter value is true.  The farther from H₀ the real parameter value is, the higher the power of any test to detect it (Sec. 6.4, skim it, takes this further)

Larger sample size gives stronger power to detect a true alternative.
Applet:  Power of a test.
(Pure decision theory:  neither hypothesis or error type is "privileged" as alpha is here)
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7.1 Inference for the mean (sigma unknown)
Need a new probability model--"Student's t" (Table D), which will compensate for not knowing sigma; then all the processes the same.
t- family:  like standard normal only slightly fatter in the tails.  Mean = 0. Symmetrical around 0.
    "Degrees of freedom" df tell which member of the t family.
      t(k) is the t distribution with k degrees of freedom.  (k is not sample size, but close.)
 Comparison with normal (Excel file)
    Lower d.f.--fatter tails.  Higher d.f.--more like standard normal.
    Table D back flyleaf: "One tail probability" (upper tail):  probability <--> "critical" t-value.