Math 151 , Fall 2004, Monday Day 37, Nov.22 Hit reload...After class

"Significance ", using table C, see Day 34
Statistically Significant result vs. convincing evidence,  important difference from H₀, see Day 33
"Significance testing" vs. "Hypothesis testing"--gathering evidence vs. making decisions. 
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 More cautions and limitations--Sec 6.3, cont'd (pp. 346-8)
>>(not in text)You cannot legitimately test a hypothesis on the same data that first suggested that hypothesis. Every data set will turn up with some unusual pattern if you examine it hard enough. 
       (If you must explore and confirm with the same data set, one way is to (randomly) take half the data set, explore and generate hypotheses; then use the other half for confirmatory tests.  You can use P-value to describe unusualness, but be wary of making decisions with it if you didn't expect that particular unusualness.)
All the warnings about designing experiments and surveys still apply.

>>Multiple Tests: beware! pp. 346-7
    If you do 100 tests and use the alpha = .05 significance level for each, then the structure of testing requires this:
    When all 100 null hypotheses H₀ are true, out of your 100, about 5 of the 100 (.05) will give "significant" results by chance alone (falsely indicating the alternative hypothesis is to be preferred.)
    Moral: if you use the testing mechanism as a screening instrument for many questions, a proportion will give falsely significant results.  You can't accept the results from such multiple tests as good evidence, only as indicating questions requiring further, more specific study. The game gives you one shot, not a hundred shots.
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Chapter 7, Inference for Distributions (we'll do 7.1, 7.2, maybe the first segment, to p. 414, of 7.3)

Inference for means, using xbar from a SRS to make inference about µ:

	Large n  Sigma known          Sigma unknown		Small n  Sigma known          Sigma unknown
normal Population is  not normal	Xbar is normal;  find z using sigma	Xbar is normal;  find z using s.	Xbar is normal;  find z using sigma	Xbar is normal;  Find t using s
	Xbar is normal-ish (CLTh);  find z using sigma	Xbar is normal-ish (CLTh);  find z using s	Unrealistic. sigma's  only "good" for  normal pop's.	(See p. 381)  If you can't use t,  Find a statistician

t-distribution family:  like standard normal only slightly fatter in the tails.  Mean = 0. Symmetrical around 0.
    "Degrees of freedom" tell which member of the t family.
      t(k) is the t distribution with k degrees of freedom.
 Comparison with normal (Excel file)
    Lower d.f.--fatter tails.  Higher d.f.--more like standard normal.
    Table C:  upper tail:  probability <--> "critical" t-value.

Start working on green box:
Assume Normal population .  Mean µ, s.d. sigma, both unknown.
Take SRS, size n, find xbar, find s (sample standard dev.)

Standard error of the (sample) mean =    Standard deviation of xbar, estimated from the data.
  "Standard error of the mean":  s/sqrt(n) SEM, SE_Xbar, etc.
        Just like sigma/sqrt(n), only s from data replaces sigma.
  When you estimate the standard deviation of a statistic,
                the resulting estimate is called the "standard error" of the statistic.

Standardizing xbar with s instead of sigma results in
   the one-sample t statistic
which has the t-distribution with n-1degrees of freedom.

We'll now repeat all the stuff from Chapter 6, only wherever there was a z, we'll substitute a t.
Here we go....
"One-sample" t- procedures: SRS of size n.  Use Xbar to estimate µ.
Substitute s for sigma in the standardizing formula. We get t instead of z, with n-1 degrees of freedom.
        It's a good idea to check for at least approximate normality in the data set.

Confidence intervals: 
   Choose t* from table C, using the n-1 row, and confidence level C.
    Special case of common pattern:    estimate + t* SE_estimate

Significance tests:  State hypotheses as in Ch. 6, find t from data, by:
 Calculating the one-sample t-statistic, using the null hypothesis value of µ (call it µ₀)
Then proceed as if it were a "z", only using the (n-1) d.f. row in  table C,
to find P-values for the t*'s it's between, write "P-value is between ___ and___".
(Or use software which will find P-value exactly. )

Example: bacteria per milliliter in 10 specimens of  raw milk from one producer.
  Parameter: actual mean bacteria/ml.
       5370, 4890, 5100, 4500, 5260, 5150, 4900, 4760, 4700, 4870

4|5  4|77 4|889  5|11  5|23

n = 10,   xbar = 4950, s = 268.45   SEM = 268.45/sqrt(10) =268.45/3.162=84.89.  deg. of freedom = 9 90% CI:  from t(9) in table, t* = 1.833   CI is 4950+1.833x268.45/sqrt(10)                                                        4950 +1.833x84.89, or  4950+155.6 bacteria/ml. If we had KNOWN Population sigma = 268.45,    we'd have used z* = 1.645, gotten a narrower CI.   (but we don't know sigma!) Test:  H0 : µ = 4800                          t = (4950 - 4800)/SEM = 150/84.89 = 1.767           Ha : µ > 4800                           t is between 1.383 and 1.833   (d.f. = 9)              (too contaminated)                P is between .10 and .05.  Some evidence for Ha (If the test had been 2-sided, P would be between .20 and .10)

SPSS--Get Handout for 7.1 Next time.  Type in above data and find P-value and CI.  (Dataset)
    Analyze>Compare Means>One-Sample T Test:  Test value = 4800.
           P-value is labeled "Sig (2-tailed)"--divide by 2 for 1 tail (if observed is in correct direction)
   Analyze>Descriptive Statistics>Explore.  Statistics button, set Confidence level.