Math 151 , Day 30, Friday, Nov. 7, 2008 hit reload....after class.

HW Day30  (Re)Read Chapter 11 (pp. 286-291 optional). First pp. 271-77   Check p. 294: 11.17, 18 (parameter/statistic, sampling dist.).  11.19, 20 (behavior of xbars, mean & s.d.)  11.22, 23, 24 (behavior of xbars, more)  Next:  Skip Ch. 12, 13.  Do Ch.14 on.
Memorize the 3 yellow-headed boxes on p. 278, 281 (mean and s.d. of sampling dist. of X-bar, Normality & Central Limit Th.)

Hand in
DISTRIBUTION OF XBAR(S) 
--These problems use only the mean and standard deviation.   
  p. 280, 11.7 (Teen cholesterol )
  p. 280, 11.8 (lab measurements)  For (b) they mean "what should n be?'

--These problems use  the "sample mean of n independent observations from a normal distribution has a normal distribution." theorem (p. 278) 
  B. Complete the problem we  worked on in class (people's temperatures) See just below the HW box.
  p. 280, 11.9 NAEP math scores  (n = 1, n = 4)
  p. 290, 11.37 and 11.39 Pollutants in auto exhausts  For 11.39:  You might want to know L so that if you tested your 25 cars and found a high value of x-bar, you would be able to compare it with L; if it was greater than L, you would go back to the manufacturer and say "I  believe you sold me a batch of bad cars, because the chances of getting an average emission level this high if the exhaust system is working properly is only 1 in 100. It is more reasonable to believe the exhaust system is not working, than that we "are" that 1 in 100 possibility."
  p. 289-90 11.36 and 11.38 Glucose testing  If we use this cutoff level L to say that people (with a mean of 4 tests) over L "have diabetes", then the chances of declaring that someone "has diabetes" when they really are OK (with mean 125mg/dl) is .05.  .05 or 5% is the chance of a "false positive" using this protocol, when the real mean is 125.

Hand in the above problems: try the ones below but keep separate, as part of the next assignment.
--These problems use the Central Limit theorem (p. 281) 
  p. 185, 11.10 What does the CLTh say?
  p. 286 , 11. 12 SAT scores, n = 1 and 70
  p. 286, 11.13, insurance (Hint: find P(Xbar> $275))
  p. 298, 11.41 auto accidents
  p. 298, 11.42 airplane overloads  (Hint: to do the problem you have to assume all the seats are taken.  Maybe not a reasonable assumption, but if there are empty seats, there's likely not a problem with overweight.)

A. If you didn't: (preliminary for Ch. 14) Get 4 slips from the Birkenstock box (outside my door if you missed class).  Record them, return them.  HW:  Find their mean xbar. Now xbar is your "point estimate" of the unknown mean of the numbers in the box.
 Calculate  xbar - .841, xbar +.841.  This is your "point estimate" plus or minus a "margin of error" of .841.  
     (xbar - .841, xbar +.841) is your "interval estimate" for the unknown mean of the box.   Be ready to add these to the list on MONDAY.

 Read, 
to discuss		 Optional 
 

B) We worked on computations using the sampling distribution of the mean. Finish these problems and complete the table below. (Sketch the density of the Xbars, label the axis, shade the desired area, for each that you do.)
"Normal" body temperature 98.6 deg. on average.  (Assume this is true.)
Assume normal distribution, & s.d.among many people is 0.6.
   Probability that one (random) healthy individual's normal temperature is above 98.8?
   Probability that the mean of a sample of 4 is above 98.8?
   Probability that the mean of a sample of 36 is above 98.8?
   Probability that the mean of a sample of 100 is above 98.8?

All of these are P(Xbar>98.8) for different sample sizes n.  (Normal table A) The answers for n = 1 and 4 are now there for you to check your work.

Sample size n
 s.d. of Xbars =  (pop.s.d.)/sqrt(n)
 z = (raw-mean)/s.d.
 P(Xbar>98.8)=  P(Z>z)
1
 .6/1 = .6
 ((98.8-98.6)/.6 = .2/.6 = .33
 		P(Z>.33) = .3707
4
 .6/2 = .3
 (98.8-98.6)/.3 = .2/.3 = .67 P(Z>.67) = .2514
36
   
100
   

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

Exams returned:  Solutions   Comments  More discussion next time
HW questions? (Day 29)

Closed book Quiz Wednesday:  Like this:  The population has mean 125 and standard deviation 18.
You take a simple random sample of size 9.  The distribution of all possible sample means from such samples has
mean _____ and standard deviation______
Answers:  Mean is 125,
 standard deviation is 18 divided by the square root of 9.   Square root of 9 is 3, so standard deviation is 18/3 = 6.
that's all.

--#11.6 SRS from a pop. of 10 grades: Add your 3 xbars to the circulating paper?
--If you didn't, Get 4 slips from the Birkenstock box.  Record them, return them (use for HW).  You'll pool the info Monday (including  4 original values)

Any more data from 10.55 and 10.56? Hand forward, please. (I can use some more numbers)

<>~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Biggest facts: What is the distribution of the random variable Xbar, when the experiment is to take a simple random sample of size n? This is the distribution of means of all possible SRS's of size n.
Call it the "sampling distribution of the (sample) mean" (p. 275-7, then details 278-86)

Today, examples, computations:
    Big facts and following details Day 29

We got as far as "Population normal --> sampling dist. of Xbar normal"
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