Advanced High-School Mathematics

386 CHAPTER 6 Inferential Statistics

its unbiased estimate s^2 x, the sample variance. We recall from
page 375 thats^2 xis defined in terms of the sample by setting

s^2 x =

1

n− 1

∑n

i=1

(xi−x)^2.

Again, this is unbiased because the expected value of this statistics is
the population varianceσ^2 (see the footnote on page 375.

We now consider the statisticT =

X−μ

Sx/

√

n

which takes on the value

t =

x−μ

sx/

√

n

from a sample of size n. Of course, we don’t know μ,

but at least we can talk about the distribution of this statistic in two
important situations, viz.,

• The sample size is small but the underlying population being sam-
  pled from is approximately normal; or


• The sample size is large (n≥30).

In either of the above two situations,T is called thetstatisticand
has what is called the t distribution with mean 0, variance 1 and
havingn−1 degrees of freedom. Ifnis large, thenT has close to a
normal distribution with mean 0 and variance 1. However, even when
nis large, one usually uses thetdistribution.^25

Below are the density functions for thetdistribution with 2 and 10
degrees of freedom (DF). As the number of degrees of freedom tends
to infinity, the density curve approaches the normal curve with mean 0
and variance 1.

(^25) Before electronic calculators were as prevalent as they are today, using thetstatistic was not
altogether convenient as thetdistribution changes slightly with each increased degree of freedom.
Thus, whenn≥30 one typically regardedTas normal and used the methods of the previous section
to compute confidence intervals. However, thetdistribution with any number of degrees of freedom
is now readily available on such calculators as those in the TI series, making unnecessary using the
normal approximation (and introducing additional error into the analyses).