Test (statistical)

See also: Test

Principle of a statistical test

The goal of a statistical test is to test an assumption concerning a whole of data.

Example

One has N achievements of a law which one knows normal (hope \ mu and variance 1), one wishes to test the assumption:
  • H_0: \ driven = 0
against:
  • H_1: \ driven \ 0

Let us calculate T_ {emp} = \ frac {\ sum_ {i=1} ^ {N+1} x_i} {\ sum (x_i- \ frac {1} {NR} \ sum_ {i=1} ^N x_i) ^2} .

Under the assumption H_0, we know the distribution of these statistics. We can then evaluate his p-been worth probability by calculating one: p_ {been worth} = P^ {H_0} \ Ge T_ {emp}

Various types of errors

In practice, the statistical tests lead to two types of errors:

  • Rejection wrongly of the assumption H_0: error of first species.

  • Acceptance wrongly of the assumption H_0: error of second species.

It is then possible to control \ alpha, the error rate of first species:

  • If p_ {been worth} < \ alpha: H_0
  • is rejected If p_ {been worth} > \ alpha: One accepts H_0
Note: In the Statistiques books it is mark that Ho Rejet if p_ {been worth} < \ alpha

According to Gujarati the p_ {been worth} is the significant level low where the perhaps rejected null assumption (translation made by my care, it may be that it is not exact to 100%) thus if p_ {been worth} > \ alpha then one does not reject

Schematically: That is to say \ alpha = 5%

If P-Been worth = 0,03:
0%---1%---2%---3%---4%---5%---6%---7%---8%---9%---10%
]… Not Rejection.
However to 5%, one rejects --> RHo

If P-Been worth = 0,05:
0%---1%---2%---3%---4%---5%---6%---7%---8%---9%---10%
] ....... Not Rejection ........
However to 5%, there is the level low which one rejects, this level is included/understood in the test --> RHO

If P-Been worth = 0,07:
0%---1%---2%---3%---4%---5%---6%---7%---8%---9%---10%
] ............ Not Rejection .............
However to 5%, one does not reject --> Non rejection of Ho (what is different from acceptance of Ho which will depend on the test of second species)

List statistical tests

Test of T

  • H0 : μ = μ0
  • H1: μ > μ0

Test of U

  • H0 : μ = μ0
  • H1: μ > μ0

Test of U

  • H0 : π = p0
  • H1: π > p0

Test of the χ2

  • H0 : σ>2 = σ02
  • H1: σ>2 > σ02

Test of U

Test of U

Test of F

Test of the χ2

Test of T

Test of Fisher-Student

Test of Spearman

Nonparametric tests

Test of Mann-Whitney

The test of (Wilcoxon-) Mann-Whitney is a nonparametric test of identity relating to two independent samples resulting from numeric or ordinal variables.

These two plays can contain numbers different of observations, or even refer to two different variables.

1) It is a test of identity: it relates to the fact that two series of values numerical (or ordinal) result from the same distribution.

2) It is nonparametric, i.e. that it does not make any assumption on the analytical forms of the distributions F1 (X) and F2 (X) of populations 1 and 2. It thus tests the assumption:

H0: " F1 = F2"

3) It not uses the values taken by the observations, but their rows once these observations joined together in the same unit.

to test an assumption concerning a whole of data.

Example

One has NR achievements of a law which one knows normal (hope μ and variance 1), one wishes to test the assumption:

The test of Mann-Whitney thus has the same objective as another important test of identity, the " Test of Chi-2 of identité" , in its version for numeric variable. If the populations are supposed to be normal and of the same variance, the test T will have the preference.

The test of Kruskal-Wallis can be perceived as an extension of the test of Mann-Whitney to more than two samples (just as univariée ANOVA is an extension of the test T to more than two samples).

Test of the sign

Test of Wilcoxon

R_ \ alpha (N) = \ frac {N (n+1)}{4} - u_ {1 \ alpha/2} \ sqrt {\ frac {1} {24} N (n+1) (2n+1)}

R_ \ alpha (N) = \ frac {N (n+1)}{4} - u_ {1 \ alpha} \ sqrt {\ frac {1} {24} N (n+1) (2n+1)}

See too

Internal bonds

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