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Probability and Statistics for Eng...

9th Edition
Jay L. Devore
Publisher: Cengage Learning
ISBN: 9781305251809

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BuyFindarrow_forward

Probability and Statistics for Eng...

9th Edition
Jay L. Devore
Publisher: Cengage Learning
ISBN: 9781305251809
Chapter 12.2, Problem 25E
Textbook Problem
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Show that b1 and b0 of expressions (12.2) and (12.3) satisfy the normal equations.

To determine

Show that b1=β^1=SxySxx and b0=β^0=y¯β^1x¯ satisfies the normal equations of Yi=β0+β1xi+εi.

Explanation of Solution

Calculation:

Least squares estimate:

In a linear equation y=b0+b1xi the constant b1 be the slope and b0 be the y-intercept form and x is the independent variable and y is the independent variable.

The error function for the equation is,

f(b0,b1)=i=1n(yi(b0+b1xi))2

Normal equations of Yi=β0+β1xi+εi:

The first normal equations of the error function will be obtained by finding the first derivative of f(b0,b1) with respect to b0 and equating it to 0.

Hence, the first normal equation is nb0+(xi)b1=yi

The second normal equations of the error function will be obtained by finding the first derivative of f(b0,b1) with respect to b1 and equating it to 0.

Hence, the second normal equation is (xi)b0+(xi2)b1=xiyi

In the linear equation the point estimates of the b0 and b1 are β^0 and β^1. These point estimates are the values that minimize the equation f(b0,b1). That is, f(β^0,β^1)f(b0,b1).

Then the values of β^0 and β^1 are the least squares estimates and the equation y^=β^0+β^1xi

Slope:

In a linear equation y^=β^0+β^1x the constant b1 be the slope and b0 be the y-intercept form and x is the independent variable and y is the independent variable.

The general formula to obtain slope is,

b1=β^1=SxySxx=(xix¯)(yiy¯)(xix¯)2

Y-intercept:

In a linear equation y^=β^0+β^1x the constant b1 be the slope and b0 be the y-intercept form and x is the independent variable and y is the independent variable.

The general formula to obtain y-intercept is,

b0=β^0=y¯β^1x¯=yiβ^1xin

Here, the objective is to prove that b1=β^1=SxySxx and b0=β^0=y¯β^1x¯ satisfies the normal equations of Yi=β0+β1xi+εi.

First normal equation:

To prove that the value nb0+(xi)b1 is yi with the substitution of b1=β^1=SxySxx and b0=β^0=y¯β^1x¯.

The computation procedure goes as follows:

nb0+(xi)b1=nβ^0+(xi)β^1=n×(y¯β^1x¯)+(xi)×SxySxx=ny¯nSxySxxx¯+n×x¯×SxySxx=ny¯

=yi

Thus, the first normal equation is satisfied by b1=β^1=SxySxx and b0=β^0=y¯β^1x¯

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Chapter 12 Solutions

Probability and Statistics for Engineering and the Sciences
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