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} $.

References

[1]	H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)
[2]	S.S. Wilks, "Mathematical statistics" , Wiley (1962)
[3]	N.V. Smirnov, I.V. Dunin-Barkovskii, "Mathematische Statistik in der Technik" , Deutsch. Verlag Wissenschaft. (1969) (Translated from Russian)
[4]	L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)
[5]	Yu.V. Linnik, "Methoden der kleinsten Quadraten in moderner Darstellung" , Deutsch. Verlag Wissenschaft. (1961) (Translated from Russian)