Transcript stats_7_2_1 NQNWU

Warm Up
1.
2.
Don’t turn in your
homework yet
The probability of getting the numbers
1,2,3,4 out of hat are 3/8, 3/8,1/8,1/8.
Construct a probability distribution (table)
for the data and a probability distribution
histogram.
a)
Find P(X>1.5)
b)
Find the largest number A for which P(X<A) <.8
Draw the density curve given by the line
segment going from (0,0) to (X,1).
a)
b)
c)
Find P(X=.5)
P(X>.5)
P (.5<X<1.5)
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AP Statistics, Section 7.2, Part 1
Warm Up
In playing a modified version of the daily 3
players select a three digit number less than
600. Such that the first digit is 0-5 and the next
two digits are 0-9. It cost $2 to play but if you
win you get $600.
1. Calculate the average winnings. (expected
payout)
2. Why is this not a fair game?
3. Calculate the standard deviation and variance
of average winnings.
4. Calculate the average profit.
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Section 7.2.1
Means and Variances
of Random Variables
AP Statistics
Random Variables: Mean
 X  p1 x1  p2 x2  p3 x3 
 X   pi xi
 pn xn
The Probabilities pi must satisfy two requirements:
1. Every Probability pi is a number between 0 and 1.
2. p1+p2+…+pn=1
 pi represents the probability of the individual event xi
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AP Statistics, Section 7.2, Part 1
Random Variables: Example
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
The Michigan Daily Game
you pick a 3 digit number
and win $500 if your number
matches the number drawn.
It costs $1 to play a single
game.
What is the average
winnings (payout) ?
 X  .001(500)  .999  0 
 .50  0
 .50
Is this a fair
game?
This tells you the average you will win per play in
the long run. (Does not take your cost into
account)
AP Statistics, Section 7.2, Part 1
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Fair Game
Average winning=cost of playing
AP Statistics, Section 7.2, Part 1
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Random Variables: Example


The Michigan Daily Game
you pick a 3 digit number
and win $500 if your
number matches the
number drawn.
What is the average
PROFIT?
 X  .001(499)  .999  1
 .499  .999
 .50
This tells you how much you will make in the long
run taking the cost into account.
Also it calculates the profit for Lotto (your negative
AP Statistics, Section 7.2, Part 1
gain is their profit)
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Random Variables: Variance

We use variance to measure the spread of a distribution
outcome
  p1  x1   x   p2  x2   x  
2
X
2
 pn  xn   x 
2
   pi  xi   x 
2
X
Mean of entire
distribution
AP Statistics, Section 7.2, Part 1
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To get the standard
deviation I would take
the square-root of the
variance.
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2
Example: Calculating variance
and standard deviation
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The Michigan Daily
Game you pick a 3
digit number and win
$500 if your number
matches the number
drawn.
The average winnings
is 0.5
 X2  (500  .5) 2  .001
  0  .5   .999
2
 249.50025  .24975
 249.75
  15.8
AP Statistics, Section 7.2, Part 1
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Using a calculator
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List 1 represents the data
 500,0
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List 2 represents the frequency of the data
 1/1000,999/1000
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STAT  CALC1-Var statsL1,L2  ENTER
AP Statistics, Section 7.1, Part 1
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Law of Large Numbers
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Samples
Draw independent observations at random from
any population with finite mean μ.
Decide how accurately you would like to
estimate μ.
As the number of observations drawn increases,
the mean x-bar of the observed values
eventually approaches the mean μ of the
population as closely as you specified and then
stays that close.
AP Statistics, Section 7.2, Part 1
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Example
The distribution of the heights of all young
women is close to the normal distribution
with mean 64.5 inches and standard
deviation 2.5 inches.
 What happens if you make larger and
larger samples…

AP Statistics, Section 7.2, Part 1
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AP Statistics, Section 7.2, Part 1
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Law of Small Numbers
Most people incorrectly believe in the law
of small numbers.
 “Runs” of heads/tails, black/red
 Something is “due”
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AP Statistics, Section 7.2, Part 1
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Assignment

Exercises:
AP Statistics, Section 7.2, Part 1
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