Regression and Correlation

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1CORRELATION & REGRESSION

1.0Introduction

Correlation and regression are concerned with measuring the linear relationship between two variables.

1.1Scattergram

It is not a graph at all, it looks at first glance like a series of dots placed haphazardly on a sheet of graph paper.

The purpose of scattergram is to illustrate diagrammatically any relationship between two variables.
(a) If the variables are related, what kind of relationship it is, linear or nonlinear ?

(b) If the relationship is linear, the scattergram will show whether it is negative or positive.

1.2Regression

It is used to identify the precise form of linear relationship between the two variables.

This is done by estimating the equation
Y = a + bx
Where Y is the dependent variable
x is the independent or explanatory variable
a is the regression constant or intercept
b is the slope or regression coefficient

In another words, it is concerned with developing the linear equation by which the value of a dependent variable Y can be estimated from a value of an independent variable.

The regression equation is most frequently found by using least square method (for which the sum of the squared deviations between the actual and estimated values of the dependent variable is minimized.)

Y = a + bx

Where b = [pic] where [pic]

a = [pic]

There are two regression lines, Y on X and X on Y . These two lines will not be unique if there is no perfect correlation. In examination questions you are usually asked to find the regression line of Y on X.

Basically, the line will be used to predict the dependent variable for a particular value of the independent variable. Thus through any bivariate distribution there will be two lines of best fit depending on which of the two variables is being considered as the independent variable.

The regression line of Y on X , Y = a + bx, is used to predict the values of Y for a given value of x.
The regression line of X on Y , X = a + bY is used to predict the values of x for a given value of Y.

If given a particular x sores, Y score could be predicted.
(a) reading from the regression line
(b)using the regression equation

Example 1
The following table shows the amount spent on advertising and the corresponding sales of the product from 10 companies

|Company |Sales £ (‘000) |Advertising cost £ (‘000) | |A |25 |8 | |B |35 |12 | |C |29 |11 | |D |24 |5 | |E |38 |14 | |F |12 |3 | |G |18 |6 | |H |27 |8 | |I |17 |4 | |J |30 |9 |

(a) Plot a scattergram showing the relationship between advertising cost and sales of the product.

(b) Calculate the equation of the regression line of sales on advertising costs.

(c) Use the regression line to forecast sales if advertising costs were £1000.

Solution:
(a)
[pic]

(b)

|Y |X |X2 |XY | |25 |8 |64 |200 | |35 |12 |144 |420 | |29 |11 |121 |319...
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