# Methods and formulas for coefficients in Fit Regression Model

Select the method or formula of your choice.

## Coefficient (Coef)

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

The formula for the intercept (b0) is:

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

b = (X'X)-1X'y

### Notation

TermDescription
yiith observed response value mean response
xiith predictor value mean predictor
Xdesign matrix
yresponse matrix

## Standard error of the coefficient (SE Coef)

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:

### Notation

TermDescription
xiith predictor value mean of the predictor
Xdesign matrix
X'transpose of the design matrix
s2mean square error

## T-value

### Notation

TermDescription test statistic for the coefficient  estimated coefficient standard error of the estimated coefficient

## P-value – Coefficients table

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:

np – 1

### Notation

TermDescription The cumulative distribution function of the t distribution with degrees of freedom equal to the degrees of freedom for error.
tjThe t statistic for the jth coefficient.
nThe number of observations in the data set.
pThe sum of the degrees of freedom for the terms. The terms do not include the constant.
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