Math 151 , Day 31, Monday, Nov.5, 2007 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
You should try the textbook problems, and bring your questions to class, and we'll work on them.  Then you'll have till Friday to hand them in.  DO do A below (with the numbers from the shoebox)
DIST. OF XBAR(S), cont'd.
These problems use  the "sample mean of n independent observations from a normal distribution has a normal distribution." theorem (p. 278) 
  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.

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. (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 WED.
 Read, 
to discuss		 Optional 
 

Exams returned:  Solutions
 Problem 0: 12 points for showing up.

problem # total
    0   1    2
    3    4    5  6      7
 8
     

possible 100 12
 8
 7
 12
 14
 21
 12
 10
 4
  9|00048
max 98
 12
 8
 7
 12
 14
 21
 12
 10
 4
  8|055678
Q3 88
 12
 6
 7
 11
 14
 19
 12
 9
 4
  7|12667799
Med 80
 12
 6
 7
 11
 12
 16
 12
 7
 3
  6|6
Q1 76
 12
 3
 7
 9 10
 13
 10
 7
 2
 
min 66
 12
 0
 0
 6
 5
 11
 8
 4
 1
 

Comments:  Reading carefully.  Knowing what words mean precisely.  Writing clearly.  Common problems: Response vs. Nonresponse bias.  Factor, values. 
HW questions? (Day 30)
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. 

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.
<>~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
RECAP: 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.
We'll call it the "sampling distribution of the (sample) mean" (p. 275-7, then details 278-86)

We talked about the above Friday.   Today, more examples, computations: Day 29, details
We worked on computations using the sampling distribution of the mean.  Will continue with those Wed.

Central Limit Theorem...
How large is "large"?  How approximate is "approximate"?
    If the population was close to normal, n doesn't need to be very large.
    If the population is not badly skewed or bimodal, n=25 already gives a pretty good approximation to normal.

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