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HW- pgs. 491-492 (7.32-7.34, 7.36)
Ch. 7 Test MONDAY 12-16-13
www.westex.org HS, Teacher Website
12-11-13
Warm up—AP Stats
Let X = # of girls a couple has in 3 tries. Find  X
(expected # of girls) and the standard deviation
of X.
# of girls
Probability
xi
pi
Σpi = 1
xipi
X =
(xi -  X )2pi
 X2 =
This can be done with our calculators:
Enter xi in L1 and pi in L2
Stat/Calc/1-var stats (L1,L2)
Calculator gives  X as x &  X .
If you want/need
variance just square  X .
Name _________________________
AP Stats
7 Random Variables
7.2 Day 2 Means and Variances of Random Variables
Date _______
Objectives
 Explain, and illustrate with an example, what is meant by the law of large numbers.
 Explain what is meant by the law of small numbers.
Statistical Estimation and the Law of Large Numbers
Let’s say we are interested in knowing the mean height µ of the _______________ of all
women between the ages of 18 and 24 years. This µ is the mean  X of the random variable X.
Since it is impossible to find the height of __________ women in the population of 18 and 24
year old women, we need to estimate  X .
So we choose an SRS of young women (18-24) and use the sample mean ___ to estimate the
unknown population mean  X . Since x is COMPUTED FROM THE SAMPLE it is called a
_______________.  X is a number that describes the POPULATION, so it is called a
_______________. **We will revisit this terminology in ch. 9, but it is HUGE for Inference
which goes from ch. 10-14.***
Statistics (such as ___) obtained from probability samples are random variables because their
values would __________ in repeated sampling. The ____________ _______________ of
statistics are just the probability distributions of these random variables. (More in ch. 9.)
The purpose of an _____ is to represent the population so it makes sense to think that ___
(the mean of our sample) should do a good job of estimating ___ (the mean of our population).
Let’s say Rachel takes an SRS of the population of 18-24 year old women and measures their
height. Nick B. does the same with his own SRS of the same size as Rachel’s. Would we expect
their sample means (___) to be exactly the same? Why not?
So if x is rarely exactly the same as  X then why is it a reasonable estimate of the population
mean (  X )? On average as our sample size increases the accuracy of x compared to the true
population mean  X increases. This is called the law of __________ numbers.
Law of Large Numbers
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 of your sample x eventually approaches the mean µ of the population as closely as you
specified.
The behavior of x is similar to the idea of probability. In the ____________, the proportion
of outcomes taking any value gets close to the probability of that value, and the average
outcome gets close to the distribution mean. Below is an example of how __________ means,
x approach the _______________ mean µ.
Thinking about the Law of Large Numbers
The law of large numbers says that the average results of many independent observations are
stable and _______________. Individual gamblers in a casino will win or lose but the casino
will win in the ____________ because the law of large numbers says what the ____________
outcome of many thousands of bets will be. Many companies besides casinos use the law of
large numbers. Can you think of some examples?
Is there a Law of Small Numbers?
The rules of probability and the law of large numbers describe the regular behavior of chance
phenomena in the ____________. Psychologists (my wife!) have discovered that our
____________ understanding of randomness is very different from the true laws of chance.
For example most people believe in the “law of small numbers” even though there is NO SUCH
LAW! People often believe that short sequences of random events will show the kind of average
behavior that appears only in the long run.
Ex. Write down a sequence of heads and tails that you think imitates 10 tosses of a fair and
balanced (like Fox News!) coin.
Gamblers often follow the hot hand theory, betting that a run will continue. Ex. I’ve won three
hands in a row at black jack, so it should keep happening. At other times they draw the
opposite conclusion. I’ve won 6 hands in a row, surely I’ll start to lose now to get back to the
expected probability which is very close to 50%. No compensation is needed to get back to the
average in the ____________. Our intuition doesn’t do a good job of distinguishing random
behavior (sometimes in the short run there will be a string of several heads or tails in a row)
from ____________ influences (something is actually wrong with the coin, so it’s not a 50-50
proposition. ***When looking at data we need statistical inference (Chs. 10-14) because
probability calculations can help verify that what we see in the data is more than a _________
pattern.***
How large is a large number?
The law of large numbers says that the actual mean outcome of many trials gets close to the
_______________ mean µ as more __________ are made. But how many trials are needed to
get close to µ? That depends on the ____________ of the random outcomes. The ________
variable the outcomes, the more trials are needed to ensure that the mean outcome ___ is close
to the distribution mean µ. Think about a person gambling at a casino. If the casino wants the
person to be interested in gambling would they want a lot of variability in the outcomes or not
too much? Now that we have decided that they would want a lot of variability (Isn’t it boring to
play a game if it’s so easy to predict what will occur?) we also know that they need _______
hands to be played to ensure that the law of large numbers comes into play.