Math 151 , Fall 2007  Mon. Day 28, Oct. 28Hit reload...after class.

HW:  Reread  Ch. 10,  Read rest (Note Normal distribution is back) . Check 10.21, 28 Read ahead, Ch. 11.
Hand in  Wednesday All.
= = = = = Probability , Ch. 10. 
p. 261, 10.16 Grades RV
p. 252, 10.6 and 10.7 D&D, 4-sided dice

Continuous sample spaces:
***For A and B, Use  Densities Handout, from Day 5. Answers to old questions 
Change language from "description of a population of data" to "pick an individual from the population, call the value X"
A. ("Uniform")   X = number the spinner points to.
a) (example)  The probability that the spinner points to a number less than .6 = P( X < .6) = .6 .
b) P (.2 < X < .6) = ?   Say it in words: ?
c) For what x is there probability .4 of being greater than x ?      (In notation: P(X > x) = .4.  Find x)
B.  Y = (number you get from) the sum of two spinners. ("Triangular")  This is the same random variable as Y in 10.14, p. 259!
a) The probability that the sum is a number less than .6  =  P(       ?        ) =.18     P(Y > .6) =  ?  
b) P(Y < 1.6)  =  ?         P (Y < 1) =   ?             P( 1 < Y < 1.6) =  ?
c)  P(Y > x) = .92  Find x:  ?   (Hint:  P(Y<x) = .08)
***
p. 259, 10.13 uniform, 0-1 (Note, this is distribution A on the handout)
p. 259, 10.14  sum of two uniform (Note, this is distribution B, "Triangular", on the handout)
pp. 236-7 10.48 and 10.50 uniform on 0-2 (This corresponds to a single spinner with its edge labeled with values going from 0 to 2, rather than the 0 to 1 we used in our homework up to now. Use the area-of-rectangles formula to find the probabilities.)

Normal distribution:  Restate each problem from "The probability that X is..." to "The proportion of the population of x's that ...." and use your old techniques.
p. 259, 10.15  Iowa Test Scores
p. 269, 10.49 Did you vote?
p. 269,  10.51 NAEP scores
p. 269,  10.53 Friends

On a separate sheet using http://www.whfreeman.com/bps4e  "Probability " applet:  If you do it jointly, one sheet for both people (I'll aggregate the results)
p. 270, 10.55 runs of free throws
p. 270, 10.56 a.  For b, do 20 people 10 times, but do 320 people only twice.  Record not only the proportion (.63 or whatever) but the fraction (like 201/320.  208/320 = .65 exactly)
 Read, to discuss


Optional 



Exams not finished; sorry!
Please add your proportion of "heads" = "0's" from 200 repetitions ("tosses") to the list and  dotplot circulating, if you didn't!
Chapter 10, Probability (intro)  continued  Details Day 24 

New today:  Random Variable Day 24 ,  and Recap  Day 25
Example of an equally likely sample space, which can be used to find probabilities for another: p.251, Ex. 10.4, 5, Two dice.
So far, everything was for Discrete sample spaces (you can list the outcomes)

Looking ahead (back)
Random variables with intervals of outcomes ("continuous") Ch.10 (p. 256 on) 
If the sample space is an interval of values (or the whole line), the way we assign probabilities to events is with a density curve (Ch. 3, cf. Day 5 on) (remember density curves were idealizations of histograms--of repeating the "experiment" many many times)
  P(a <  X < b) = the probability that X is between a and b  is the area under the density curve, between a and b.
We declare P (X = a) = 0 , so P(a <  X < b) = P(a < X < b)
Notation: Use capital letter for the random variable, the "label" of the phenomenon.  Use small letters for particular values it can have.  But this rule is often broken--Moore uses x-bar where many (I) would use X-bar.
  HW: For A and B, Use Densities Handout, from Day 5. Answers to old questions 
Change language from "description of a population of data" to "pick an individual from the population, call the value X"
A. ("Uniform")   X = number the spinner points to.
a) (example)  The probability that the spinner points to a number less than .6 = P( X < .6) = .6 .
b) P (.2 < X < .6) = ?   Say it in words: ?
c) For what x is there probability .4 of being greater than x ?      (In notation: P(X > x) = .4.  Find x)
B.  Y = (number you get from) the sum of two spinners. ("Triangular")  This is the same random variable as Y in 10.14, p. 259!
a) The probability that the sum is a number less than .6  =  P(       ?        ) =.18     P(Y > .6) =  ?  
b) P(Y < 1.6)  =  ?         P (Y < 1) =   ?             P( 1 < Y < 1.6) =  ?
c)  P(Y > x) = .92  Find x:  ?   (Hint:  P(Y<x) = .08)
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Our most important probability model: NORMAL DISTRIBUTION family.  Same techniques as before, only we ask "probability that one chosen at random..." instead of "proportion of all..."  Review Normal techniques: Day 8 covers it all.
 Take a random sample of size 1 from a population which is N(110, 25). 
(Give an individual, chosen at random, the "Classic IQ test", which has a normal distribution, mean 110, s.d. 25.   X is the score on the test.)
Find P(100 < X < 145), prob. that individual gets between 100 and 145. (what we did in class)   Work is on Day 8, what proportion.

Teaser for Ch. 11:  We know that a sample from a population will not exactly represent the population.  If we take a random sample, the behavior of samples will not be individually predictable, but there will be predictable pattern in many random samples from the same population.  Knowing the pattern will be  as good as we can do.   Remember the Random Sample of 6 "minority" managers, #8.10 How many East Asians did you get in a sample of size 6?  Discussion
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