Math 151 , Spring 2005, Day 30 Friday, April 15 Hit reload After class

Day 30 (Friday, April 15): Reading: Ch. 19, Confidence Intervals for Proportions.  Ch. 20 next.
Hand in (All D&V)
For next class: if you didn't finish finding the 68% and 95% CI's for your sample, please do, and be ready to add your results to our list:  
# of 1's, p-hat, SE(p-hat), p-hat + SE, ME for 95% = 1.96SE, p-hat + 1.96SE

Postpone handing in all the rest:  Do the green parts tonight and hang on to them:
Ch. 19p. 386
5, 6 Conclusions. Do these with the "Don't misstate..." section, pp. 361-2.
9 Cars
A.   Use the Normal table to find z* for a 99% CI (This is the z* for which 99% of the area is between -z* and +z*).  Find z* for a 90% CI. (Text p.358 shows that z* = 1.645 to 3 decimal places.  Normal table only gives 2 places.)  Check your results by comparing with the corresponding results in Table T p. A-53.
11 Ghosts  Do the computations for part a tonight.
13 Teenage drivers  You can do all of this tonight.
21 Rickets
3, 4 Conditions
16 Local news

ME, C, n pp. 356-7, 361-3.  Problems p. 368
7, 8  Relationships
23 Deer ticks
25 Graduation  The answers in the back use the 25% as the p to plug in.  Redo part a (only) using 50% as the p (what you would do if you had no idea what p would be.). How many subjects do you "save" by using the 25%?
26 Hiring
28 Hiring again
29  Pilot study

 Read,
  to 
discuss 		 Optional 
Homework questions? Day 29
Got through definition and computation of CI:  next time, introduce table T, and Assumptions/conditions, continue with material below.
Level C confidence interval for population proportion p: 
 "One -proportion  z-interval"   Chapter 19  Day 29

Note Trade-off:  Higher Confidence ---Wider interval (bigger ME. Less "precision")
Desire:  Small Margin of Error + High confidence.  p. 361-2


But they grow and shrink together: High confidence--Low precision ; High precision (small ME)--low confidence.

Way out:  increase n, the sample size.  (Shrinks SE)  How big a sample size for desired ME and C?
   Plan ahead:  Decide on desired ME and C(thus z*).  Guesstimate p (p=1/2 requres largest sample size--safest).
Solve equation for n.   (Some results pre-calculated, p. 362)
Notes:  --To cut ME in half, need 4 times the sample size.  Certainty/precision are expensive!
    -- If you're sure your p will be far from 1/2,  you can get a smaller n by using a closer guesstimate for p.

Green shoebox:  To get a 90% CI, ME = .04:  use p = 1/2 = .5.    z* = 1.645.
          n = (1.6452) ( .5· .5) / (.042 )  =  2.706025 · .25/.0016= .67650625/.0016 = 422.8  Round UP! to 423.

Why does it work??  Why does the ME calculated this way give intervals that capture the real p C% of the time??
   Think about the Sampling distribution of p-hat.  It's Normal, center at the real (population) p. SD(p-hat) is its standard deviation. SE(p-hat) approximates SD(p-hat)
Now ME = z*SE(p-hat),  where + z* cut off the center C% of the standard normal model.
So, in the Sampling distribution model, Real p+ ME  spans the center C% of this normal curve.
So the probability that p-hat falls in the range Real p + ME  is C%; That is, with many random samples, the proportion of p-hats that fall in the range Real p + ME  is C%.
That is, the proportion of p-hats that are within the distance ME of p---is C%

Now:  If p-hat is within ME of p, then p is within ME of p-hat.  The "arms" (+ ME ) that a p-hat interval sticks out from p-hat will capture p, if and only if p-hat is within ME of p.  But the proportion of p-hats that do that is C%.


 
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