| Hand in
Continuous sample spaces: Do the questions A and B in webpage below, with the Tables for simple models (densities) handout Normal model: Restate each problem using: "The probability that x is..." ~ ~ "The proportion of the population of x's that ...." and use your old techniques. p. 352 #21 a, b only. Pregnancy Find the solutions, then Restate a and b from proportion questions to probability questions: "What is the probability that a pregnancy chosen at random will last... ... " and "If the chances of a random pregnancy lasting longer than k days is.... ..., then what is k?" p. 265 #28a only Rivets p. 353 #33a only IQ's, #34a only Milk Probability, Ch. 15. p. 299
A. Newly assigned after class, for Day 26: On separate page: Use the "Probability" Applet at http://bcs.whfreeman.com/scc to simulate tossing a penny 25 times. Write down h, the number of heads, and p-hat, the proportion of heads you got (p-hat = h/25). Reset, and repeat, till you have a total of 10 simulations. Make a dotplot (by hand; see D&W p. 36) of your 10 p-hats, using an axis marked with .28, .32, .36, .40, .44, .48, .52, .56, .60, .64 etc. (add numbers at the ends if you need them). Bring your list and dotplot to class to hand in and to share. (Be glad I didn't have you flip a real penny.) On separate paper: Start the rest; read
them all. Do what you can. Will be assigned next time.
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Read,
to discuss |
Optional |
P(a < x < b) = the probability
that the outcome x is between a and b
is the area under the model's density curve, between a and b.
is the proportion of x's which would come up between a and b
if we did the phenomenon a zillion times.
We declare P(a) = 0 (In a continuous model, getting precisely
a
is utterly unlikely; can't even measure that well),
so P(a <
x
< b) = P(a < x < b)
Review ""Tables for simple models (densities)""
HW day 7, restating these parts as probability
questions:
(Copies of the HW handout
are outside my door if you can't find yours.)
Change language from "description of a population of data" or
"area between/above/below" to
"pick an individual at random the population, call the
value x"
A. ("Uniform") Spin the spinner once.
x = number the spinner points to.
a) (example) The probability that the spinner points to a number
less than .6 = P( x < .6) = area to left of .6 = .6.
b) P (.2 < x < .6) = ? Say it
in words: ?
c) For what c is there probability .4 of being greater than
c
?
(In notation: P(x > c) = .4. Find c)
B. y = (number you get from) the sum
of two spinners.
("Triangular")
a) The probability that the sum is a number less than .6 =
P( ?
) =.18
b) P(y > 1.6) = ? P(y < 1.6)
= ? P (y <
1) = ?
P( 1 < y < 1.6) = ? Say each in probability
words.
c) P(c > x) = .08. Find c: ? Say
in words.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- - - - - -
Our most important probability model: NORMAL Model family.
Same techniques as before, only we ask "probability that one chosen at
random..." instead of "proportion of all..." Review Normal techniques:
Day
8, Day 9
Take a random sample of size 1 from a population which is
N(110, 25) = =
Give an individual adult, chosen at random, the Wechsler test, which
has a normal distribution, mean 110, s.d. 25. x is her score
on the test.
Find P(100 < x < 140), prob. that the individual chosen
at random gets between 100 and 140. Same as: of all individuals,
fraction who score between 100 and 140. Work is on Day
9, what proportion.
= = = = = = = = = = = = = = = = = =
Back to Chapter 16, Probability rules
Equally likely probabilities: If there are k possible
outcomes, and each is equally likely, each has probability 1/k (Dice, coin,
etc. Plenty of things are not equally likely.)
(Remember:) If A and B are disjoint events, then P( A or B) =
P(A) + P(B)
If A and B can happen together, P(A and B) is not 0.
Then P(A or B) = P(A) + P(B) - P(A and B).
"General addition rule."
Use Venn diagram to see. Example
A thousand people are interviewed by the census bureau, and the results
tabulated in this two way table.
Working Status vs. Sex.
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Restated as proportions of the whole:
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Two variables: Restate conditional frequency distributions
as conditional probabilities. Recall Day
3
P(B|
A): Probability of B "given"
A:
Probability that B happens, conditional
on knowing that A happens.
When you write or see percents, be clear what
is on the bottom of the fraction (even if it takes longer to
say)!!.
Start here Friday
Marginal distribution: Distribution of one variable, ignoring/summingover
the other.
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P ( Woman)= 50% = .5, P( In Labor
Force) = 80% = .8.
Conditional distribution: Distribution of one variable,
with the individuals being only those which satisfy a condition in the
other variable.
Choose a person: Conditional distribution
of working status given that a woman was chosen.
Dist. of working status given that a man was
chosen.
| Women | Men | Total | |
| In Labor Force | 350/500 = 70% | 450/500 = 90% | 80% |
| Not in Labor Force | 150/500 = 30% | 50/500 = 10% | 20% |
| Total | 500/500=100% | 500/500=100% | 100% |
P( In Labor Force | Woman)
= 350/500 = .7 P (In Labor Force | Man) = 450/500 = .9
Given that the person chosen is in the labor
force, conditional distribution as to
sex.
Given that the person chosen
is not in the labor force, conditional distribution as to sex.
"Row
%s"--rows add to 100%: "conditional distributions of sex by working
status."
| Women | Men | Total | |
| In Labor Force | 350/800 = 43.8% | 450/800 = 56.2% | 800/800=100% |
| Not in Labor Force | 150/200 = 75% | 50/200 = 25% | 200/200=100% |
| Total | 50% | 50% | 100% |
Another formula: P(B|
A) = P(Band
A)/P( A)
P( Woman | In Labor Force ) = P( Woman and In Labor
Force )/ P( In Labor Force )
= (350/1000) ÷ (800/1000) = .35/.80 = 350/800 = .438
Independence (definition) A and B are independent
if - and only if- P(B|
A) = P(B)
Knowing A is true (or not) makes no difference
to the probability of B.
A being true (or not) does not change the chances for
B.
Now if A and B are independent, then P(B|
A) = P(B) . But P(B|
A) = P(Band
A)/P( A) .
So P(B) = P(Band
A)/P( A) . Multiply both
sides by P( A).
Get
P(Band
A) = P(B) × P(
A) This was our Rule 5 from p. 275 (Day
23, bottom)
IF Sex and Working Status were independent,
the table would look like this: (or close
to it)
Working Status vs. Sex.
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Restated as proportions of the whole:
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| Sievers home | Math151-Sp05/Days26.htm | 1pm | 4/6/05 |