Applied Statistics and Probability for Engineers
Applied Statistics and Probability for Engineers
6th Edition
ISBN: 9781118539712
Author: Douglas C. Montgomery
Publisher: WILEY
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Chapter 11, Problem 96SE

a.

To determine

Show the expression i=1n(yiy^i)=0  is correct.

a.

Expert Solution
Check Mark

Explanation of Solution

Given info:

It is given that Y is the response variable and x is the predictor variable.

Calculation:

Let, Y be the response variable and x be the predictor variable, then the simple linear regression model can be expressed as Y=β0+β1x+ε, where, β0 and β1 are the intercept and slope coefficient respectively. ε is the random error with mean zero and variance σ2.

From the linear regression it is known that,

yi=β0+β1xi+εiy^i=β^0+β^1xi

From the normal equation,i=1nyi=nβ^0+β^1×i=1n(xi)

i=1ny^i=i=1n(β^0+β^1xi)=i=1nβ^0+β^1×i=1n(xi)=nβ^0+β^1×i=1n(xi)

Substituting the values,

i=1n(yiy^i)=i=1nyii=1ny^i=nβ^0+β^1×i=1n(xi)nβ^0+β^1×i=1n(xi)=0

Hence, the expression i=1n(yiy^i)=0 is correct.

b.

To determine

Show the expression i=1n(yiy^i)xi=0  is correct.

b.

Expert Solution
Check Mark

Explanation of Solution

Calculation:

From the linear regression it is known that,

yi=β0+β1xi+εiy^i=β^0+β^1xi

i=1n(yiy^i)xi=i=1nyixii=1ny^ixi

From the normal equation,i=1nxiyi=β^0i=1n(xi)+β^1×i=1n(xi2)

i=1ny^ixi=i=1n(β^0+β^1xi)xi=i=1nβ^0xi+β^1×i=1n(x2i)=β^0i=1nxi+β^1×i=1n(x2i)

Substituting the values,

i=1n(yiy^i)xi=i=1nyixii=1ny^ixi=β^0i=1n(xi)+β^1×i=1n(xi2)β^0i=1nxiβ^1×i=1n(x2i)=0

Hence, the expression i=1n(yiy^i)xi=0 is correct.

c.

To determine

Show the expression 1ni=1ny^i=y¯  is correct.

c.

Expert Solution
Check Mark

Explanation of Solution

Calculation:

From the linear regression it is known that,

y^i=β^0+β^1xiβ^0=y¯β^1x¯

i=1ny^i=i=1n(β^0+β^1xi)=i=1nβ^0+β^1×i=1n(xi)=nβ^0+β^1×i=1n(xi)

Substituting the values,

1ni=1ny^i=nβ^0+β^1×i=1n(xi)n=n(y¯β^1x¯)+β^1×i=1n(xi)n=ny¯nβ^1x¯+β^1nx¯n=ny¯n=y¯

Hence, the expression 1ni=1ny^i=y¯ is correct.

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Chapter 11 Solutions

Applied Statistics and Probability for Engineers

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