statistics: inferential 483

The central-limit theorem states that, for many dif-

ferent samples of 1,500 people, the statistic ˆ

pwill vary

in value according to a normal distribution with mean

pand standard deviation where, in this

case, N= 1,500. As the value of Nis large, the stan-

dard deviation is small, meaning that all values of ˆ

p

will be closely clustered about the mean value p. In

particular, this establishes, as we would expect, that ˆ

p=

45 percent is a good estimate for p. The key is to now

ask, How good?

From the study of the

NORMAL DISTRIBUTION

, the

68–95–99.7 rule states that 95 percent of the mea-

surements for ˆ

pfall within a distance of two standard

deviations from the mean p. That is, there is a 95 per-

cent chance that our measurement of ˆ

p= 45 percent

lies within the range of values to

(This range of values also contains

p, of course, at its center.)

As an approximation, we substitute into these for-

mulae the value ˆ

p= 45 percent for p:

This yields a range of values [42.4, 47.6] that, in this

approximation, and with approximately a 95 percent

level of confidence, contains the true proportion value p.

We call this range of values a 95% confidence interval. If

these calculations were performed on a large number of

survey results (all involving 1,500 people) then we

would be sure that close to 95 percent of the intervals

produced contain the true population proportion p.

2. Estimating a Population Mean:

In a media study 680 young adults, ages 21 to

25 years, were given a test on current events.

Scores on the test ranged from 0 to 500, indi-

cating a range of knowledge on the topic. The

mean score was m= 170 (and the standard

deviation was 80). On the basis of this sample,

what can be said about the mean knowledge

level (score) µof the population of all 19 mil-

lion young adults?

The central-limit theorem states that, for many dif-

ferent samples of 680 young adults taking the test, the

mean score mwill vary in value according to a normal

distribution with mean µand standard deviation .

Here σis the standard deviation for the entire popula-

tion (unknown) and Nis the sample size (N= 680).

Since the value of Nis large, the standard deviation

will be small. This means two things: that the mean

m= 170 is likely to be close to the true mean value µ,

and that using the standard deviation of 80 observed in

the sample as an approximation for the true value σ

will not seriously alter our calculations. With this said,

the 68-95-99.7 rule states that there is a 95 percent

chance that our observed value m= 170 falls within

two standard deviations of the true mean value µ. As

an approximation, then, we evaluate:

yielding a 95 percent confidence interval of [163.9,

176.1] for what would be the mean score if the entire

population were to take the test.

In 1908 W

ILLIAM

S

EALY

G

OSSET

(1876–1937), pub-

lishing under the pseudonym “Student,” made a more

precise analysis of the distribution of mean values from

normal distributions. If ˆ

σis the standard deviation of a

sample of size N, Gosset calculated the distribution of

values for the sample mean mone would expect using ˆ

σ

as an approximation for the true standard deviation σ

of the population. In particular, he described the distri-

bution of the

Z

-

SCORE

of the mean m: