MATH 251, Probability and Statistics I, Fall 2001, Fri. Nov. 16, Day 33 corrected

Exam 2: Open book takehome.  Available today.   Due in class Wednesday Nov. 28 (Day 36)
Cleaning up odds and ends from Significance testing:
--Multiple tests will produce spuriously significant results. If a test is significant al level .05, call that a "success".  The probability of "success" for a single test, where H0 is true, is .05.  If you do tests on a bunch of questions (where all the H0's are true) you're doing a Binomial situation.
--Two-sided test, level alpha, can be done by finding level (1-alpha) CI.
Significance testing vs. hypothesis testing  (gathering evidence vs. making decision)
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Chapter 7, Inference for Distributions (we'll do 7.1, 7.2, and the first segment, to p. 567, of 7.3)
        GOSSETT HANDOUT
Inference for means, using xbar from a SRS:
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 If you can't use t (see p. 516),
see pp. 518-22 or 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.
    Lower d.f.--fatter tails.  Higher d.f.--more like standard normal.
    Table D:  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" = s/sqrt(n)    Standard deviation of xbar, estimated from the data.

Standardizing xbar with s instead of sigma results in
t =    xbar -µ
         s/sqrt(n)         the one-sample t statistic
which has the t-distribution with n-1 degrees of freedom.

We'll now repeat all the stuff from Chapter 6, only wherever there was a z, we'll substitute a t.
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Quiz

HW Day 33 

Hand in: 
Sec. 6.3 
p. 482 , 6.60 Bonferroni procedure  If you do multiple tests as a fishing expedition, individual items that come up significant at (e.g.) .05 could have happened by chance, or could indicate "real" differences from their null hypotheses.  Then you would need to make a new study, collect new data, to check on these.  If you have no possibility of this, and need conclusions from this  set of data, you can use this procedure.  It is analogous to dividing the alpha (here .05)  between two tails for a two sided test--here we divide the alpha evenly among all the separate tests--anything that still comes up significant when it is that far out can legitimately be deemed significant at the .05 level overall.   What do we lose?  Power!  The chances of confirming any real difference from the null hypothesis go down, by demanding to be farther out in the tail. 

6.62  probability in multiple tests  This is like p.451, p. 619.
You can do it using the normal approximation, or using the formula: prob that 2 or more are sig = 1- prob (0 or 1 are sig.)
(You can also use SPSS: pp. 76-7 in your SPSS book.)


Sec. 7.1
Problems 1,2,3,4 on back of Gosset handout.		 Read, discuss
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 

 Optional
(more practice)
Sec. 6.3, p. 483 6.61 (if you want one to check the back of the book with)
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