# Compression Analysis Of Linear Regression And Correlation

*Register to read the introduction…* The relationship between the variables is linear. Both variables must be at least interval scale. The least squares criterion is used to determine the equation.

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Regression Analysis – Least Squares Principle

The least squares principle is used to obtain a and b. The equations to determine a and b are:

n( XY ) ( X )( Y ) b n( X 2 ) ( X ) 2 Y X a b n n

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Illustration of the Least Squares Regression Principle

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Regression Equation - Example

Recall the example involving Copier Sales of America. The sales manager gathered information on the number of sales calls made and the number of copiers sold for a random sample of 10 sales representatives. Use the least squares method to determine a linear equation to express the relationship between the two variables. What is the expected number of copiers sold by a representative who made 20 calls?

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Finding the Regression Equation - Example

The regression equation is : Y a bX Y 18.9476 1.1842 X Y 18.9476 1.1842(20) Y

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The means of these normal distributions of Y values all lie on the straight line of regression. The standard deviations of these normal distributions are equal. The Y values are statistically independent. This means that in the selection of a sample, the Y values chosen for a particular X value do not depend on the Y values for any other X values.

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Confidence Interval and Prediction Interval Estimates of Y

•A confidence interval reports the mean value of Y for a given X. •A prediction interval reports the range of values of Y for a particular value of X.

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Confidence Interval Estimate - Example

We return to the Copier Sales of America illustration. Determine a 95 percent confidence interval for all sales representatives who make 25 calls.

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Confidence Interval Estimate - Example

Step 1 – Compute the point estimate of Y In other words, determine the number of copiers we expect a sales representative to sell if he or she makes 25 calls.

The regression equation is : Y 18.9476 1.1842 X Y 18.9476 1.1842(25)

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^ ^

Y 48.5526