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One of the basic methods for statistical hypotheses testing (cf. [[Statistical hypotheses, verification of|Statistical hypotheses, verification of]]), used to confront the outcomes of observations  $  x _ {1} \dots x _ {n} $, 
interpreted as realizations of random variables  $  X _ {1} \dots X _ {n} $, 
with a hypothesis  $  H _ {0} $. 
It is usually based on the observed value of a suitable statistic  $  T = T ( X _ {1} \dots X _ {n} ) $, 
the specific form of which depends on the formulation of the problem. Generally speaking, the application of a significance test does not presuppose the availability of an alternative hypothesis  $  H _ {1} $, 
to be accepted if  $  H _ {0} $
is rejected; however, if some  $  H _ {1} $
is specified, then, according to the general theory of statistical hypotheses testing, it is precisely  $  H _ {1} $
that determines the choice of the statistic  $  T $, 
the governing principle being to maximize the power of the test (cf. [[Power of a statistical test|Power of a statistical test]]).

Significance tests are usually applied as follows. Having selected the test by choosing in advance a statistic  $  T ( X _ {1} \dots X _ {n} ) $
and a significance level  $  \alpha $, 
$  0 < \alpha < 0.5 $, 
determine the [[Critical value|critical value]]  $  t _  \alpha  $
of the test by the condition  $  P \{ T \geq  t _  \alpha  \mid  H _ {0} \} = \alpha $. 
According to the significance test with level  $  \alpha $, 
the hypothesis  $  H _ {0} $
is rejected if  $  T ( x _ {1} \dots x _ {n} ) \geq  t _  \alpha  $. 
If  $  T ( x _ {1} \dots x _ {n} ) < t _  \alpha  $, 
the conclusion is that  $  H _ {0} $
is not in disagreement with the observations  $  x _ {1} \dots x _ {n} $, 
at least as long as new observations do not oblige the experimenter to adopt a different point of view.

Example. A counter monitoring a [[Poisson process|Poisson process]] registers 150 pulses during the first hour, 117 pulses during the second. Can it be assumed on this basis that the pulse arrival rate per unit time is constant (hypothesis  $  H _ {0} $)?

If  $  H _ {0} $
is true, the observed values 150, 117 may be treated as realizations of two independent random variables  $  X _ {1} $
and  $  X _ {2} $
obeying the same Poisson law with parameter  $  \lambda $; 
the value of  $  \lambda $
is unknown. Since  $  H _ {0} $
implies that the random variables

$$ 
Y _ {1}  =  \sqrt {4 X _ {1} + 1 } - 2 \sqrt \lambda \  
\textrm{ and } \  Y _ {2}  = \ 
\sqrt {4 X _ {2} + 1 } - 2 \sqrt \lambda
$$

are approximately normally distributed with parameters  $  ( 0 , 1) $, 
it follows that the statistic

$$ 
X  ^ {2}  =  
\frac{1}{2}
 [ Y _ {1} - Y _ {2} ]  ^ {2}  =  
\frac{1}{2}
 [ \sqrt {4 X _ {1} + 1 } - \sqrt {4 X _ {2} + 1 } ]  ^ {2}
$$

is approximately distributed according to a  "chi-squared"  law with one degree of freedom, i.e.

$$ 
{\mathsf P} \{ X  ^ {2} > x \mid  H _ {0} \}  \approx  {\mathsf P} \{ \chi _ {1}  ^ {2}
> x \} .
$$

Tables of the  "chi-squared"  distribution yield the critical value  $  \chi _ {1}  ^ {2} ( 0.05 ) = 3.841 $
for the prescribed significance level  $  \alpha = 0.05 $, 
i.e.

$$ 
{\mathsf P} \{ \chi _ {1}  ^ {2} \geq  3.841 \}  =  0.05 .
$$

Now, using the observed values  $  X _ {1} = 150 $
and  $  X _ {2} = 117 $, 
one computes the value  $  X  ^ {2} $
of the test statistic:

$$ 
X  ^ {2}  =  
\frac{1}{2}
 [ \sqrt {4 \cdot 150 + 1 } - \sqrt {4 \cdot 117 + 1 } ]  ^ {2}  =  4.087 .
$$

Since  $  X  ^ {2} = 4.085 > \chi _ {1}  ^ {2} ( 0.05 ) = 3.841 $, 
it follows that the hypothesis  $  H _ {0} $(
constant pulse arrival rate) is rejected by the  "chi-squared"  test at significance level  $  \alpha = 0.05 $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cramér,   "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.L. Lehmann,   "Testing statistical hypotheses" , Wiley  (1959)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.V. Smirnov,   I.V. Dunin-Barkovskii,   "Mathematische Statistik in der Technik" , Deutsch. Verlag Wissenschaft.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.I. Devyatov,   "Limits of admissibility of normal approximations to the Poisson distribution"  ''Theor. Probab. Appl.'' , '''14''' :  1  (1969)  pp. 170–173  ''Teoriya Veroyatnost. i ee Primenen.'' , '''14''' :  1  (1969)  pp. 175–178</TD></TR></table>