Math 151 , Fall 2005, Day 37 Mon. Nov. 21  Hit reload ..

Exam not finished.
Buffer against one low hour exam:
The final % exam grade minus 10 points will be substituted for the lowest hour exam grade, if it is higher.
Examples:
Ex1 Ex2 Ex3 final % final -10
Student 1 Original 90 80 60 85 75, replaces lower 60
Treated 90 80 75 85 ß These will be used.
Student 2 Original 90 80 70 75 65, lower than 70, don't replace.
Treated 90 80 70 75
Student 3 Original 90 50 55 85 75, replaces lower 50
Treated 90 75 55 85 ßThese will be used

This is to encourage those who have had trouble to try to put it together for the final.


Day 37: (Re)Reading: Chapter 20+21 thru p. 392 (Activstats is good here too.) (exam to here.) Then continue (Alpha levels) through 395.  Lightly through Error types and Power . Read What can go wrong p. 401 and the rest. (SPSS won't do proportion computations, but some other programs do; it's good to have an idea what you might see, p. 402.) Next: Chapter 23, Means.
Hand in (All D&V)  Monday after break!

Two-sided:  For some reason, D&V don't model or assign any 2-sided problems (except #8).  We need to be used to them for later, so here are a few.
b) Use your green shoebox result to do a Two sided test against the null hypothesis p = .5.
Ch. 20, p.387  #8 Find the mistakes
From ActivStats, copied here:
 MRA-304-2:  Kerrich Coin Toss  While he was a prisoner of the Germans during World War II, the British statistician John Kerrich tossed a coin 10,000 times.  He got 5067 heads.  Take Kerrich's tosses to be an SRS from the population of all possible tosses of his coin.  If the coin is perfectly balanced, p = 0.5.  Is there reason to think that Kerrich's coin was not balanced? 

<> TRE-396-9:  Store Checkout-Scanner Accuracy (adapted from Activstats HW):
In a study of store checkout-scanners, 1234 items were checked and 20 of them were found to be overcharges (based on data from "UPC Scanner Pricing Systems: Are They Accurate?" by Goodstein, Journal of Marketing, Vol. 58).  Before scanners were used, the overcharge rate was estimated to be about 1% . Based on these results, do scanners appear to give a different rate of overcharges than the old method of keying in the price?  (All items had to have individual price tags; scanning is much less labor-intensive.)  Do the steps, finding the P-value and stating a conclusion. 
= = = = = = = = = = 
"Significance" Ch. 21, p. 404 
1 P-value
3, 4 Alpha 
5, 6 Significant?
+ + + + + + + + + + +
A.  Use the T-table to decide these questions: 
a)  Ho: p = .3 vs.  HA: p>.3.   z from p-hat is 2.12.  Is it significant at the .01 level? .05? .10? 
b)  Ho: p = .3 vs.  HA: p not = .3.   z from p-hat is 2.12.  Is it significant at the .01 level? .05? .10? 
c)  Ho: p = .3 vs.  HA: p>.3.   z from p-hat is 3.16.  Is it significant at the .01 level? .05? .10? 
p. 387, #11 (use p.397--CI's & Tests)
Read,
  to 
discuss 		 Optional
 
Error type & power:
p. 404, #7, #13
Continuing with Hypothesis testing (often called Significance testing)

Use CI to estimate true value.  Two-sided tests.    Notes:  Day 32

"Statistically significant" result, and  "alpha"  "significance level."    Cautions.  Notes Day 34
More about alphas:
Especially if we must make a decision to Reject Ho  (or retain it)---
  Set "benchmark" or "cutoff" level  "alpha"  "significance level":  (p. 393-4)
        If  P-value is less than alpha, we say the test is "significant at level alpha"
                      (Seeing the result (again) would be rarer than alpha, if the null hypothesis is true)
Table T (A-53)  bottom row is z-values.

What if you don't have the Z-table but only have the T-table (Table p. A-53)?
What if you have a demanded level of significance, alpha?
"Critical value" --the z* corresponding to your alpha (p.394-5 )
   T-Table: a limited list of probabilities  across the top row:
            = Right tail values for the bell curve distribution.  (and double that for equal-tails)
        The value in the bottom (infinity or z*) row under the probability is the corresponding standard normal value.
        "z* is the upper p critical value of the standard normal distribution."
  Do this: Find your z from the data. Make a sketch of the normal curve and mark z on it.  Mark the direction(s) of Ha.
    (If your z is in the direction of Ha , continue.  Otherwise the results are hopelessly not significant: you can quit.)
Find the two z*'s in Table T (p. A-53) that bracket your z (ignore minus sign).  Find the corresponding p's.
    e.g. z =1.83

Two tail p         .10     .05      .02
One tail p   ...   .05     .025     .01 ...
infinity(z*)      1.645 \/ 1.960    2.326
                    z = 1.83

Notice as the z's increase, the amounts in the tail(s) decrease.
Test is significant at the bigger bracketing probability; not sig. at the smaller. For z = 1.83,
One sided: P-value is less than .05 and greater than .025
       Significant at the .05 level,not at the .025 level
Two sided: P-value is less than .10 and greater than .05
       Significant at the .10 level,not at the .05 level
If you have a specific demanded significance level, compare it with these levels.
Give P-values if you can! (more information)

Confidence Intervals and Hypothesis Tests: (p. 397)
 Suppose you're interested in  Ho: p = .3 vs.  HA: p not = .3  ( two-sided alternative).  If your 95% CI for p DOES NOT include the po value (.3) , then you can Reject Ho at the .05 level (.05 = 1.00 -.95).  This is approximate, because we use different calculations for the standard deviations, but good enough if the CI is not close to the po.

More about decisions in testing.
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