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Key Equations

(4-1)

Y = β_{0} + β_{1} X + P

$Y = β_{0} + β_{1} X + P$

Underlying linear model for simple linear regression.

(4-2)

\hat{Y} = b_{0} + b_{1} X

$\hat{Y} = b_{0} + b_{1} X$

Simple linear regression model computed from a sample.

(4-3)

e = Y - \hat{Y}

$e = Y - \hat{Y}$

Error in regression model.

(4-4)

b_{1} = \frac{\sum (X - \bar{X}) (Y - \bar{Y})}{\sum {(X - \bar{X})}^{2}}

$b_{1} = \frac{\sum (X - \bar{X}) (Y - \bar{Y})}{\sum {(X - \bar{X})}^{2}}$

Slope in the regression line.

(4-5)

b_{0} = \bar{Y} - b_{1} \bar{X}

$b_{0} = \bar{Y} - b_{1} \bar{X}$

Intercept in the regression line.

(4-6)

SST = Σ {(Y - \bar{Y})}^{2}

$SST = Σ {(Y - \bar{Y})}^{2}$

Total sums of squares.

(4-7)

SSE = Σ e^{2} = Σ {(Y - \hat{Y})}^{2}

$SSE = Σ e^{2} = Σ {(Y - \hat{Y})}^{2}$

Sum of squares due to error.

(4-8)

SSR = Σ {(\hat{Y} - \bar{Y})}^{2}

$SSR = Σ {(\hat{Y} - \bar{Y})}^{2}$

Sum of squares due to regression.

(4-9)

SST = SSR + SSE

$SST = SSR + SSE$

Relationship among sums of squares in regression.

(4-10)

r^{2} = \frac{SSR}{SST} = 1 - \frac{SSE}{SST}

$r^{2} = \frac{SSR}{SST} = 1 - \frac{SSE}{SST}$

Coefficient of determination.

(4-11)

r = \pm \sqrt{r^{2}}

$r = \pm \sqrt{r^{2}}$

Coefficient of correlation. This has the same sign as the slope.

(4-12)

s^{2} = MSE = \frac{SSE}{n - k - 1}

$s^{2} = MSE = \frac{SSE}{n - k - 1}$

An estimate of the variance of the errors in regression; n is the sample size and k is the number of independent variables.

(4-13)

s = \sqrt{MSE}

$s = \sqrt{MSE}$

An estimate of the standard deviation of the regression. Also called the standard error of the estimate.

(4-14)

MSR = \frac{SSR}{k}

$MSR = \frac{SSR}{k}$

Mean square regression. k is the number of independent variables.

(4-15)

F = \frac{MSR}{MSE}

$F = \frac{MSR}{MSE}$

F statistic used to test significance of overall regression model.

(4-16)

Y = β_{0} + β_{1} X_{1} + β_{2} X_{2} + \dots + β_{k} X_{k} + ϵ

$Y = β_{0} + β_{1} X_{1} + β_{2} X_{2} + \dots + β_{k} X_{k} + ϵ$

Underlying model for multiple regression model.

(4-17)

\hat{Y} = b_{0} + b_{1} X_{1} + b_{2} X_{2} + \dots + b_{k} X_{k}

$\hat{Y} = b_{0} + b_{1} X_{1} + b_{2} X_{2} + \dots + b_{k} X_{k}$

Multiple regression model computed from a sample.

(4-18)

Adjusted r^{2} = 1 - \frac{SSE/ (n - k - 1)}{SST/ (n - 1)}

$Adjusted r^{2} = 1 - \frac{SSE/ (n - k - 1)}{SST/ (n - 1)}$

Adjusted $r^{2}$ $r^{2}$ used in building multiple regression models.

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Table of Contents for Key Equations

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Table of Contents for
Key Equations