AP Statistics Section 7.2 B
Law of Large Numbers
We would like to estimate the
mean height,  ,of the population
of all American women between
the ages of 18 and 24 years.
To estimate  , we choose an SRS of
young women and use the sample
mean, x, as our best estimate of  .
Recall that a statistic is a value
obtained from a sample
_____, while a
parameter is a value obtained from
a population
________.
Statistics, such as the mean,
obtained from probability samples
are random variables because their
values would vary in repeated
sampling.
The sampling distribution of a statistic
is just the probability distribution of
the random variable.
We will discuss sampling distributions
in detail in Chapter 9.
Is it reasonable to use x to estimate  ?
It depends!!
We don’t expect x   and we realize
that x will probably change from one
SRS to the next. So what could we do
to increase the reasonableness of
using x to estimate  ?
Choose a larger sample size
This idea is called the The Law of Large
Numbers, which says, broadly anyway, that
as the SRS increases, the mean x of the
observed values eventually approaches the
mean,  , of the population and then stays
close.
Casinos, fast-food restaurants and
insurance companies rely on this
law to ensure steady profits.
Many people incorrectly believe in
the “law of small numbers” (i.e.
they expect short term behavior to
show the same randomness as
long term behavior). This is
illustrated by the following
experiment.
Write down a sequence of heads
and tails that you think imitates 10
tosses of a balanced coin.
__ __ __ __ __ __ __ __ __ __
How long is your longest string (called a
run) of consecutive heads or tails? _____
Most people will write a sequence with no
more than _____
2 consecutive heads or
tails. Longer runs don’t seem “random” to
us.
The probability of a run of three or
more consecutive heads or tail in
10 tosses is actually greater than
____.
.5 This result seems surprising
to us. This result occurs in sports as
well with the idea of a “hot hand”
in basketball or a “hot bat” in
baseball.
Careful study suggests that runs of
baskets made or missed are no
more frequent in basketball than
would be expected if each shot
was independent of the player’s
previous shots. Gamblers also
follow the hot-hand theory, also to
no avail.
Remember that it is only in the
long run that the regularity
described by probability and the
law of large numbers takes over.