Select the method or formula of your choice.

The formula for the coefficient or slope in simple linear regression is:

The formula for the intercept (*b*_{0}) is:

In matrix terms, the formula that calculates the vector of coefficients in multiple regression is:

**b** = (**X'X**)^{-1}**X'y**

Term | Description |
---|---|

y_{i} | i^{th} observed response value |

mean response | |

x_{i} | i^{th} predictor value |

mean predictor | |

X | design matrix |

y | response matrix |

For simple linear regression, the standard error of the coefficient is:

The standard errors of the coefficients for multiple regression are the square roots of the diagonal elements of this matrix:

Term | Description |
---|---|

x_{i} | i^{th} predictor value |

mean of the predictor | |

X | design matrix |

X' | transpose of the design matrix |

s^{2} | mean square error |

Term | Description |
---|---|

test statistic for the coefficient | |

estimated coefficient | |

standard error of the estimated coefficient |

The two-sided p-value for the null hypothesis that a regression coefficient equals 0 is:

The degrees of freedom are the degrees of freedom for error, as follows:

*n* – *p* – 1

Term | Description |
---|---|

The cumulative distribution function of the t distribution with degrees of freedom equal to the degrees of freedom for error. | |

t_{j} | The t statistic for the j^{th} coefficient. |

n | The number of observations in the data set. |

p | The sum of the degrees of freedom for the terms. The terms do not include the constant. |