# Student test

$t$- test

A significance test for the mean value of a normal distribution.

## The single-sample Student test.

Let the independent random variables $X _ {1} \dots X _ {n}$ be subject to the normal law $N _ {1} ( a, \sigma ^ {2} )$, the parameters $a$ and $\sigma ^ {2}$ of which are unknown, and let a simple hypothesis $H _ {0}$: $a = a _ {0}$ be tested against the composite alternative $H _ {1}$: $a \neq a _ {0}$. In solving this problem, a Student test is used, based on the statistic

$$t _ {n-} 1 = \sqrt n \frac{\overline{X}\; - a _ {0} }{s} ,$$

where

$$\overline{X}\; = \frac{1}{n} \sum _ { i= } 1 ^ { n } X _ {i} \ \textrm{ and } \ \ s ^ {2} = \frac{1}{n-} 1 \sum _ { i= } 1 ^ { n } ( X _ {i} - \overline{X}\; ) ^ {2}$$

are estimators of the parameters $a$ and $\sigma ^ {2}$, calculated with respect to the sample $X _ {1} \dots X _ {n}$. When $H _ {0}$ is correct, the statistic $t _ {n-} 1$ is subject to the Student distribution with $f = n- 1$ degrees of freedom, i.e.

$${\mathsf P} \{ | t _ {n-} 1 | < t \mid H _ {0} \} = \ 2S _ {n-} 1 ( t) - 1,\ \ t > 0,$$

where $S _ {f} ( t)$ is the Student distribution function with $f$ degrees of freedom. According to the single-sample Student test with significance level $\alpha$, $0 < \alpha < 0.5$, the hypothesis $H _ {0}$ must be accepted if

$$\left | \sqrt n \frac{\overline{X}\; - a _ {0} }{s} \right | < t _ {n-} 1 \left ( 1 - \frac \alpha {2} \right ) ,$$

where $t _ {n-} 1 ( 1- \alpha /2)$ is the quantile of level $1- \alpha /2$ of the Student distribution with $f= n- 1$ degrees of freedom, i.e. $t _ {n-} 1 ( 1- \alpha /2)$ is the solution of the equation $S _ {n-} 1 ( t) = 1- \alpha /2$. On the other hand, if

$$\left | \sqrt n \frac{\overline{X}\; - a _ {0} }{s} \right | \geq t _ {n-} 1 \left ( 1 - \frac \alpha {2} \right ) ,$$

then, according to the Student test of level $\alpha$, the tested hypothesis $H _ {0}$: $a = a _ {0}$ has to be rejected, and the alternative hypothesis $H _ {1}$: $a \neq a _ {0}$ has to be accepted.

## The two-sample Student test.

Let $X _ {1} \dots X _ {n}$ and $Y _ {1} \dots Y _ {m}$ be mutually independent normally-distributed random variables with the same unknown variance $\sigma ^ {2}$, and let

$${\mathsf E} X _ {1} = \dots = {\mathsf E} X _ {n} = a _ {1} ,$$

$${\mathsf E} Y _ {1} = \dots = {\mathsf E} Y _ {m} = a _ {2} ,$$

where the parameters $a _ {1}$ and $a _ {2}$ are also unknown (it is often said that there are two independent normal samples). Moreover, let the hypothesis $H _ {0}$: $a _ {1} = a _ {2}$ be tested against the alternative $H _ {1}$: $a _ {1} \neq a _ {2}$. In this instance, both hypotheses are composite. Using the observations $X _ {1} \dots X _ {n}$ and $Y _ {1} \dots Y _ {m}$ it is possible to calculate the estimators

$$\overline{X}\; = \frac{1}{n} \sum _ { i= } 1 ^ { n } X _ {i} \ \textrm{ and } \ \ \overline{Y}\; = \frac{1}{m} \sum _ { j= } 1 ^ { m } Y _ {j}$$

for the unknown mathematical expectations $a _ {1}$ and $a _ {2}$, as well as the estimators

$$s _ {1} ^ {2} = \frac{1}{n-} 1 \sum _ { i= } 1 ^ { n } ( X _ {i} - \overline{X}\; ) ^ {2} \ \textrm{ and } \ \ s _ {2} ^ {2} = \frac{1}{m-} 1 \sum _ { j= } 1 ^ { m } ( Y _ {j} - \overline{Y}\; ) ^ {2}$$

for the unknown variance $\sigma ^ {2}$. Moreover, let

$$s ^ {2} = \frac{1}{n+} m- 2 [( n- 1) s _ {1} ^ {2} + ( m- 1) s _ {2} ^ {2} ].$$

Then, when $H _ {0}$ is correct, the statistic

$$t _ {n+} m- 2 = \frac{\overline{X}\; - \overline{Y}\; }{s \sqrt 1/n+ 1/m }$$

is subject to the Student distribution with $f = n+ m- 2$ degrees of freedom. This fact forms the basis of the two-sample Student test for testing $H _ {0}$ against $H _ {1}$. According to the two-sample Student test of level $\alpha$, $0 < \alpha < 0.5$, the hypothesis $H _ {0}$ is accepted if

$$| t _ {n+} m- 2 | < t _ {n+} m- 2 \left ( 1 - \frac \alpha {2} \right ) ,$$

where $t _ {n+} m- 2 ( 1- \alpha /2)$ is the quantile of level $1- \alpha /2$ of the Student distribution with $f= n+ m- 2$ degrees of freedom. If

$$| t _ {n+} m- 2 | \geq t _ {n+} m- 2 \left ( 1- \frac \alpha {2} \right ) ,$$

then, according to the Student test of level $\alpha$, the hypothesis $H _ {0}$ is rejected in favour of $H _ {1}$.

How to Cite This Entry:
Student test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Student_test&oldid=48883
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article