Math 151 , Day 21, Friday, March 16, 2001

            Exam 2 on  Day 24, the Friday of the week we return.
Ch. 4, Probability and Sampling Distributions.
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.

Sec. 4.1:
                  Population  Choose from it a Sample (varies)
Calculate
Numerical summary:  Parameter(Greek letter)    Statistic (Latin)
    Examples:               Population mean mu                Sample mean xbar
                            Pop. standard dev. sigma         Sample st. dev. s
                          Pop. median                    Sample median
                          Pop. proportion p              Sample proportion p-hat

The actual value of the Statistic will vary,depending on the particular sample. "Sampling variability"
The Statistic "estimates" the Parameter.  We hope it is close to the parameter.  If we choose simple random samples, we can understand the pattern of values the statistic can take.

Chance  behavior (a random phenomenon): Unpredictable in the short run,  predictable regular pattern in the long run.
 "Probability" of particular something happening: proportion of times it would happen in a very long series of independent repetitions of the phenomenon.
    (independence:  outcome of one trial (repetition) must not influence the outcome of any other.)

Some sample statistics:
    Height:   U.S. young women: pop. mean= 64.5", pop. s.d. 2.5"  (text p.66.  Caveat: rounded?)
                                                       This class:, xbar = 64.2,  s = 3.75
    Coin flip: Proportion of heads  p = 1/2 (?)       p-hat =  256/520 = .492
    Thumbtack:  Proportion of point-up p =  (??)       p-hat =  441/691 = .6382
    Spinning penny*: Proportion of heads p =  (??)   1999 pooled   p-hat =  321/650 = .4938
                                                                                      This year's? 398/750 = .531

Sampling                          .   .
variability                       .   .
Distribution of                   .   .
12 p-hats, 1999, + 15, 2001       .   .
Proportion of heads/50        .   .   .
(sec. 4.3)                    .   .   . 
                              .   .   . 
      .       .   .   .   .   .   .   .       .       .
--+---+---+---+---+---+---+---+---+---+---+---+---+---+-     
 .36 .38 .40 .42 .44 .46 .48 .50 .52 .54 .56 .58 .60 .62

Prof. Persi Diaconis (a table magician) can flip a coin so precisely it always comes up the way he wants.  His coinflipping is not a random phenomenon.  Mine is.


Sec. 4.2 Probability Models (will be done Monday after break)
Random phenomenon,
    Sample space S:  set of all possible outcomes
    Event:  any outcome or set of outcomes
    Probability model: S, and a way of assigning a probability to each event.
Sample space depends on what you want to know:  Flip coin twice.
    S1 = {HH, HT, TH, TT}     S2 = {0, 1, 2} number of heads   S3 = {Y, N} both heads?

Probability rules:  pp. 222-3, in words, then in notation.
A an event in sample space S, P(A) is "the probability of A"
    These rules are all true for proportions in long run, prop.of counts, proportions of areas.
    1.  0 < P(A) < 1
    2. P(S) = 1
    3. For any event A, P(A does not occur) = 1 - P(A)
    4.  A and B are  disjoint if they have no outcomes in common (can't happen simultaneously.)
        If A and B are disjoint, their probabilities add:  P(A or B) = P(A) + P(B)

Pick one person from U.S. Pop. (Age 25 +)
Sample space:
No HS degree
       HS only     .
1-3 yrs College
 4 + yrs College
Proportion in pop.
18.3%
33.9%
24.8%
23.0%
Probability 
.183
.339
.248
.230
P(not finished college) =
P( HS or less) =

HW Day21, Friday, March 16  final version Read  4.1, 2  Next: Read ahead, 4.3.  Skip 4.4 and Skip Ch. 5.
Hand in For Day 21
p. 215,  4.1, 2, 3 parameter/statistic
= = = = = = = =

These will be assigned Monday (Read 4.2, to p. 27, 4.3, pp.236-38) 
Probability 
p. 2.18 4.6 random digits
 4.9 3 of a kind
 4.10 numbers-->words
 4.12 world series prob?
Sec. 4.2 Probability models: 
p. 221  4.14 sample spaces
p. 224  4.16 social mobility in Denmark
 4.17 cause of death
 4.18 husbands' share
Finite sample spaces 
p. 226  4.19 legitimate dice?
  4.21 
p. 232  4.31 SRS size 2
4.32 farm size
Random variable language--finite sample spaces still 
p. 231 4.25 sum of 2 dice
p. 235 4.35 social mobility in England
= = = = = = = = = 
Section 4.3, thru p. 238
p. 238 4.38 law of large numbers		 Read, to discuss 
 
 
 
 
 
 
 
 
 
 

 

 Optional 

= = = = = =


 
 
 

(more of same) 
4.15sample spaces
 
 

4.28 land in Canada
4.29 m&m

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*Text answer book says spinning pennies often approach 40% heads in the long term.  Not here. Are Wells Women different?!