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, , as our best estimate of .

x

Recall that a statistic is a value obtained from a _____, while a

parameter is a value obtained from a ________.

sample

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 to estimate ?

We don’t expect and we realize that will probably change from one SRS to the next. So what could we do

to increase the reasonableness of using to estimate ?

x

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xx

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size samplelarger a Choose

This idea is called the The Law of Large Numbers, which says, broadly anyway, that

as the SRS increases, the mean of the observed values eventually approaches the mean, , of the population and then stays

close.

x

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 _____ consecutive heads or

tails. Longer runs don’t seem “random” to us.

2

The probability of a run of three or more consecutive heads or tail in 10 tosses is actually greater than ____. 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.

5.

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.