Math 151 , Day 26, Wednesday, April 3, 2002after class

- HW questions? Continuous Random Variables.  Normal Random Variables. Most of class was spent on this.
-From # 4.40 p.241  Add your 3 means to the circulating list.  If you didn't add yours today, add them Friday.

SPSS simulation: average of  spinners which can land on any number between 0 and 1.
  Population--one spinner.  distribution flat between 0 and 1, mean .49 s.d. = .29
  n = 2, Average of 2 spinners is Xbar.  Distribution triangular between 0 and 1, mean .50, s.d. .21.  .29/sqrt(2) =.205
  n = 4, Average of 4 spinners is Xbar.  Distribution normalish between 0 and 1, mean .50, s.d. .15.  .29/sqrt(4) =.145
  n = 15, Average of 15 spinners is Xbar.  Distribution normal between 0 and 1, mean .50, s.d. .09.  .29/sqrt(15) =.076

Mean of Xbars is mean of population.
Standard deviation of Xbars is s.d. of population divided by square root of n.
As sample size increases, sampling distribution of Xbars gets more and more normal-shaped.
   (Central Limit Theorem)

 Friday we will continue looking at the Central Limit Theorem, starting here, then begin Ch. 6.
Sec. 4.3: Sampling distributions. We take a simple random sample of size n, find the sample mean xbar.  It will be different depending on the sample, so we have a random phenomenon.  We measure the outcome as a number, the sample mean, so we have a random variable Xbar.
Law of large numbers  Let the sample size n get bigger.  Then  the xbars will eventually get very close to the population mean mu.

Usually we have a fixed sample size n.  Assume that's true from now on.
What probability distribution describes the random phenomenon of finding Xbar from a SRS?
That is, what is the distribution of the random variable Xbar, when the experiment is to take a simple random sample of size n?
We'll call it the "sampling distribution of the sample mean" = distribution of means of all possible SRS's of size n.
  Shape, center, spread, (outliers?)

Look at results from #4.40.

Things we know:

 Example: "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  individual's  normal temperature is below 98.0 degrees?
       Take SRS of 9 people.  Sampling distribution of the mean?  Probability that the mean is below 98.0?
   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?

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.
    Even if the population is pretty weird, n=25 gives a pretty good approximation to normal.
Pictures on overhead.


PreClass assignment Day 26 for  Day 27
Activstats: DO Ch 16 if you didn't.  P. 16-3 is the same as 16-2 with an added activity, + some new comments.
For next classes. Ch. 17 (optional)  introduces Confidence intervals. (Alternative, read Moore 6.1)  It can give you some good reinforcement for what we are doing.  But many instructors (me included) think his  demos with ybar in the center of the distribution may give a misleading sense of the definition of confidence interval (tho his words are right.).  So if you work with this chapter, be sure to do p. 17-4 (Appendix) after p. 17-1.

If YOU DIDN'T HAVE IT WEDNESDAY, DO Moore sec. 4.3 Sampling dist. of the sample mean:
p. 241, 4.40 do a and b; Do b this way.  Close your eyes and put your finger down somewhere on table B.  Start reading the table where your fingertip lands.  Find xbar for your sample
Repeat part b, to get a total of 3 values of xbar. (You can just keep reading the table where you left off, or you can put your finger in a different spot).  Make a dotplot of your 3 values and bring the values to class to be compiled with everyone else's.
HW Day26 Read sec. 4.3. (Skip ch. 5) Sec. 6.1 next.  Read ahead...
Memorize the tan box on p. 242 (mean and s.d. of sampling dist. of x-bar)
Do all of these as part of Day 27, to be handed in Wednesday. OOPS Monday
     Read, try now is a good idea.
These problems use only the mean and standard deviation. 
  p. 243, 4.41 (lab measurements)
  p. 250, 4.50
These problems use either the Central Limit theorem, or the "sample mean of n independent observations from a normal distribution has a normal distribution." theorem (both on p. 244)
  p. 249, 4.51 cola (you did a, now do b) 
  p. 247, 4.44 carpet flaws.  Also draw some square yards and mark some flaws.
  p. 250, 4.53 auto accidents
More problems:
p. 243 4.42 unbiasedness, sample size
p. 249 4.52 hypokalemia
p. 249 4.48 dust Note, the dust actually weighs 123mg, but the weighings may not be accurate enough for us to find the actual weight. "Distribution of this mean" = "Distribution of means from all possible sets of 3 weighings from these scales." When I took physics, we did not have digital scales; they were balance beams; and we weighed everything 3 times and found the average. (Have you ever gotten on the scale, said "that can't be right!" gotten off and on again a couple times?) 
. 
p.250 4.54 (labeled 4.53?)pollutants; backward from value to probability.  You might want to know L so that if you tested your 125 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 hit that 1 in 100 possibility." 		 Read, to discuss Optional
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