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Essentials of Statistics for the B...

8th Edition
Frederick J Gravetter + 1 other
ISBN: 9781133956570

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Chapter
Section
BuyFindarrow_forward

Essentials of Statistics for the B...

8th Edition
Frederick J Gravetter + 1 other
ISBN: 9781133956570
Textbook Problem

For the following set of data,

  1. a. Find the linear regression equation for predicting Y from X.
  2. b. Calculate the standard error of estimate for the regression equation.
X Y
7 6
9 6
6 3
12 5
9 6
5 4

a.

To determine

To Find: The regression equation for predicting Y from the X values.

Answer

The regression equation for predicting Y from the X values is Y^=0.25X+3_.

Explanation

Given info:

The data represents the values of X and Y.

Calculation:

The general regression equation for predicting Y from the X values is,

Y^=bX+a

Where, parameter b is calculated as b=SPSSx and parameter a calculated as a=MybMx

The formula for SP is as follows:

SP=(XMx)(YMy)

The formulae for squared deviations SSx and SSy are as follows:

SSx=(XMx)2 and SSy=(YMy)2

The below table showing the calculations required for calculating correlation:

S.no.XYDeviationDeviation square Product
(XMx) (YMy) (XMx)2 (YMy)2 (XMx)(YMy)
1 7 6 -1 1 1 1 -1
2 9 6 1 1 1 1 1
3 6 3 -2 -2 4 4 4
4 12 5 4 0 16 0 0
5 9 6 1 1 1 1 1
6 5 4 -3 -1 9 1 3

Sum/

average

8 5     32 8 8

So, the sum of product of deviation (SP) is 8, SSx is 32 and SSy is 8. Now substitute the above calculated values in the b value then:

b=SPSSx=832=0.25

Now substitute the above calculated values in a value then:

a=MybMx=50.25×8=52=3

So, a value is calculated as 3 and b value is calculated as 2. Now substitute these values in the general regression equation then:

Y^=bX+a=0.25×X+3=0.25X+3

Thus, the regression equation is Y^=0.25X+3_.

b.

To determine

To calculate: The standard error of estimate for the equation.

Answer

The standard error of estimate for the equation is 1.22.

Explanation

Calculation:

The Pearson correlation (r) is calculated as:

r=SP(SSx)×(SSy)

Substitute the above calculated values in the correlation formula then:

r=SP(SSx)×(SSy)=8(32)×(8)=816=0.5

So, the Pearson correlation is 0.5.

The SS residual also known as unpredicted variability is calculated as:

SSresidual=(1r2)SSy

Where r is calculated above as 0.5 as r2 is the squared correlation, so r2 is 0.25 and SSy is calculated above as 8. Substitute these values in the SSresidual formula then:

SSresidual=(1r2)SSy=(10.25)×8=0.75×8=6

So, SSresidual is computed as 6.

The standard error of estimate is calculated as:

SE=SSresidualdf

Here SE is for standard error of estimate and d.f is degree of freedom calculated as:

df=n2=62=4

Now substitute the above calculated values in the standard error of estimate formula then:

SE=SSresidualdf=64=1.5=1.22

Thus, the standard error of estimate is 1.22.

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