482 statistics: inferential

example, or grams if they are weights) rather

than units squared, as is required for variance.

In a

NORMAL DISTRIBUTION

approximately 68 per-

cent of the data values lie within one standard devia-

tion of the mean (either side), 95 percent within two

standard deviations, and 99.7 percent within three.

This known as the 68–95–99.7 rule. It shows that

standard deviation does indeed give a good indication

of how widely the data values in a distribution are

scattered. A small standard deviation, for example,

indicates that 68 percent of the data values are closely

clustered about the mean.

See also

DISTRIBUTION

;

PERCENTILE

;

SCATTER DIA

-

GRAM

;

STATISTICS

:

INFERENTIAL

.

statistics: inferential The science of drawing general

conclusions about a population based solely on numer-

ical information gathered from a sample of that popu-

lation is called inferential statistics. (See

POPULATION

AND SAMPLE

.) For example, a medical study might

observe that 16 of 100 army inductees have blood type

AB. One might then infer that approximately one-sixth

of the world’s entire population is of this blood type.

An assertion or conjecture about a numerical fea-

ture of a population is called a hypothesis. For exam-

ple, the assertion “one-sixth of the world’s population

is of blood type AB” is a hypothesis that seems to be

supported by the medical study described above. It is

unclear, however, whether another study of 600 college

seniors, 79 of whom were of blood type AB, also sup-

ports the claim.

A statistical test is a mathematical procedure that

allows one to determine, to some specified degree of

confidence, whether or not the results of a particular

study support a hypothesis. The claim being tested for

acceptance or rejection is called the null hypothesis. The

alternative hypothesis is the assertion to be accepted if

the null hypothesis is deemed false.

The principles of statistical testing are well summa-

rized within the following example:

Suppose one is given a coin. We wish to deter-

mine whether or not the coin is fair, that is,

whether tossing a head is just as likely as toss-

ing a tail. We take as the null hypothesis the

statement, “The coin is fair,” and alternative

hypothesis, “The coin is biased.” To test the

hypothesis let us say we toss the coin 10 times.

Suppose we obtained 10 heads in a row. What

should we conclude?

As the chances of tossing 10 heads in a row with a

fair coin are very small—the probability of this occurring

is —we would perhaps conclude

that the coin is biased. But there is a small chance that a fair

coin could have nonetheless produced this result. To reflect

this degree of uncertainty, we can say that we come to the

conclusion that the coin is biased with a “99.9 percent

level of confidence.” The statistical test performed here

was a probability calculation. We came to accept the null

hypothesis with a 99.9 percent level of confidence.

The rejection of a null hypothesis when in fact it

was true is known as a type I error (maybe the coin

was fair). If the null hypothesis is accepted despite

being false, a type II error is committed. A level of con-

fidence (or significance level) of a statistical test is the

probability of committing a type I error. A 95 percent

level of confidence is generally deemed acceptable. (As

a side note, the chances of tossing nine heads among a

series of 10 tosses are . If, in

our experiment, this is what we observed, then we

would conclude again that the coin is biased, with a 99

percent level of confidence. Observing eight heads

among a series of 10 tosses leads to the same conclu-

sion with a 95.6 percent level of confidence.)

The

CENTRAL

-

LIMIT THEOREM

provides the means

for performing statistical tests useful for analyzing pub-

lic surveys and polls. We present here two examples

illustrating two common applications.

1. Estimating Population Proportion:

In a recent Gallup poll 1,500 people were sur-

veyed, and 45 percent of them agreed that taxes

on gasoline should be raised. To what extent

does this figure represent the true proportion of

all people in this country who hold this view?

Let prepresent the true (but unknown) percentage

of Americans who believe that gasoline taxes should be

raised. Our task is to find the value of p. All we have to

work with is the observation that one sample of 1,500

people produced a proportion ˆ

pof 45 percent holding

this view.