# Distribution Tests

## Chi-Square Goodness of Fit Test

##### Description

Determines if the sample derives from a population with a normal distribution the mean and variance of which are calculated from the sample itself.

$ H_{0}: $ Data in sample derives from a population with a normal distribution

$ H_{1}: $ Data in sample does not derive from a population with a normal distribution

##### Returns

- p-value
- decision
- Chi-Sq statistic

## Jarque Bera Test

##### Description

Uses the sample skewness and sample kurtosis to determine if the sample derives from a normal distribution of unknown mean and variance.

$ H_{0}: $ Data in sample derives from a population with a normal distribution

$ H_{1}: $ Data in sample does not derive from a population with a normal distribution

##### Returns

- p-value
- decision
- JB statistic

## One Sample Kolmogorov Smirnov Test (KS1)

##### Description

Determines if the sample derives from a population with a standard normal distribution.

$ H_{0}: $ Population CDF of the sample is the same as std normal CDF

Tail | Alt. Hypothesis |
---|---|

$\scriptstyle smaller\hspace{1mm}$ | Population CDF of the sample is smaller than std normal CDF |

$\scriptstyle larger\hspace{1mm}$ | Population CDF of the sample is larger than std normal CDF |

$\scriptstyle unequal\hspace{1mm}$ | Population CDF of the sample is not same as std normal CDF |

##### Returns

- p-value
- decision
- KS statistic

## Two Sample Kolmogorov Smirnov Test (KS2)

##### Description

Determine if the two samples derive from the same distribution but make no assumptions about what that distribution is.

$ H_{0}: \text{population CDF of the sample 1 is the same as population CDF of sample 2} $

Tail | Alt. Hypothesis |
---|---|

$\scriptstyle smaller\hspace{1mm}$ | Population CDF of the sample 1 is smaller than population CDF of sample 2 |

$\scriptstyle larger\hspace{1mm}$ | Population CDF of the sample 1 is larger than population CDF of sample 2 |

$\scriptstyle unequal\hspace{1mm}$ | Population CDF of the sample 1 is not same as population CDF of sample 2 |

##### Returns

- p-value
- decision
- KS statistic