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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 scores,

X Y
3 6
6 1
3 4
3 3
5 1
  1. a. Find the regression equation for predicting Y from X.
  2. b. Calculate the predicted Y value for each X.

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^=1.25X+8_.

Explanation

Given info:

The data represents the scores 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

Regression equation:

SP=(XMx)(YMy)

Squared deviations SSx and SSy are calculated as:

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 3 6 -1 3 1 9 -3
2 6 1 2 -2 4 4 -4
3 3 4 -1 1 1 1 -1
4 3 3 -1 0 1 0 0
5 5 1 1 -2 1 4 -2

Sum/

average

4 3     8 18 -10

So, the sum of product of deviation (SP) is 10, SSx is 8 and SSy is 18.

The value of regression coefficient is,

b=SPSSx=108=1.25

The value of the intercept of the regression equation is,

a=MybMx=3(1.25)×4=3+5=8

The regression equation is,

Y^=bX+a=(5/4)×X+8=1.25X+8

Thus, the regression equation is Y^=1.25X+8_.

b.

To determine

To find: The predicted Y values from the given X values.

Answer

The predicted Y values for the given X scores are:

X Y Predicted Y value
3 6 4.25
6 1 0.50
3 4 4.25
3 3 4.25
5 1 1.75
Explanation

Calculation:

The regression equation for predicting Y from the X values is Y^=1.25X+8.

Substitute the given X scores in the above calculated regression equation for getting the predicted Y values. The predicted Y values are showing in the below table:

XYPredicted Y value ( Y^=1.25X+8)
3 6 4.25
6 1 0.50
3 4 4.25
3 3 4.25
5 1 1.75

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