From previous sections we learn that for random samples from a normal population with the mean μ and the variance σ2, the random variable \( \overline{x} \) has a normal distribution with the mean μ and the variance \( \frac{\sigma^2}{n} ; \) in other words,
\[ \dfrac{\overline{x} - \mu}{\sigma/\sqrt{n}} \] has the standard normal distribution. This is a very important result, but the major difficulty in applying it is that in most realistic applications the population standard deviation σ is unknown. This makes it necessary to replace σ with an estimate, usually with the value of the sample standard deviation s.
If Y and Z are independent random variables, Y has a
chi-square distribution with ν degrees of freedom, and Z has the
standard normal distribution, then the distribution of
\[
T = \frac{Z}{\sqrt{Y/\nu}}
\]
is given by
\[
f(t) = \dfrac{\Gamma \left( \frac{\nu +1}{2} \right)}{\sqrt{\pi\nu} \,\Gamma
(\nu /2)} \left( 1 + \frac{t^2}{\nu} \right)^{-1-\nu /2} \qquad\mbox{for }
- \infty < y < \infty,
\]
and it is called the t distribution> with ν degrees of
freedom.
The t distribution was introduced originally by the English statistician William Sealy Gosset (1876--1937), who published his scientific writings under the pen name "Student," since the brewery company for which he worked did not permit publication by employees. Thus, the t distribution is also known as the student-t distribution, or student's t distribution.
Theorem: If \( \overline{x} \) and s are the
mean and standard deviation of a random sample of size n from a
normally distributed population with mean μ, then
\[
T = \frac{\overline{x} - \mu}{s/\sqrt{n}}
\]
has t distribution with n-1 degrees of freedom. ■
We build t-distribution from normal distributions: