Regression Corr

8/2/2019 Regression Corr

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Simple Linear Regression & Correlation

We now consider bivariate data. Bivariate data is data

connecting two variables:

The independent or explanatory variable, and

The dependent or response variable.


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A scatter diagram is used to display bivariate data.

We are interested if there is a linear relationship between

the two sets of data displayed. This can fall in one of 3

classes:

1. POSITIVE LINEAR CORRELATION

2. NEGATIVE LINEAR CORRELATION

3. NO LINEAR CORRELATION


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Scatter diagrams show plots of ordered pairs .

Usually, represents the independent variable and

represents the dependent variable. The following

scatter diagrams show different classes of linear

relationships:

x

y

, x y


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Graph showing positive correlation


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Graph showing negative correlation


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Graph showing no linear correlation


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A scatter diagram is a good indication as to whether the

linear relationship between the two sets of data can be

represented by a mathematical equation.

This mathematical equation is called a regression function.

We can determine the equation of the regression line

representing the proposed relationship.


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Simple Linear Regression & Correlation The least squares regression line of on :

The general equation of a straight line is

where represents the gradient and represents the

y-intercept. The regression equation is very similar:

The regression equation takes the form

where will represent the gradient and will represent the

y-intercept. These values are called the regression coefficients.

y

y mx c

x

cm

y a bx

ab


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These values for and are determined by the following

formulae: and

where and are the mean of and the mean of

respectively. The formula for is based on the fact that

MUST be a point on the regression line.

N.B. and

22n xy x

b n x x

a y bx

a b

x y x

y , x y

a

n

x x

n

y y


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A way to measure the strength of a linear relationship between two

variables is using the product moment correlation coefficient .

This is a number which lies between -1 and +1 . Consider 3 cases:

If then there is positive linear correlation.

If then there is negative linear correlation.

If then there is no linear correlation present

r

0r

0r

0r


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The correlation coefficient is calculated using the formula:

From this formula we can also determine the coefficient of determination :

This measure represents the ratio of explained variation to total variation.

2 22 2

nxy xyr

nx x ny y

2

2 22 2

nxy xy

cdr nx x ny y


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*A Practical ExampleTabulated is the maintenance cost/00s$ to the age/mths of 10 X -ray machines.

i. Find the least squares regression line of maintenance cost on age and use it to

predict the maintenance cost for a similar machine 40 mths old.

ii. Calculate the product moment correlation coefficient between the age of the

machine and the cost for maintenance.

iii. Determine the percentage variation in the total maintenance cost that is explained

by the variation in the machine age.

Question adapted from Business Mathematics & Statistics

395500000351000504090ost ( y )

6000000050ge ( x )10achine


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From the table on the previous slide we know that:

i. Using the formulae from slide 9

The regression equation is

Estimated cost =

2 210; 300; 2970; 97650; 12050;

nxy xy x y

2 22

10 97650 300 29702.8

10 12050 300

2970 3002.8 21.

10 10

nxy x y

b nx x

y xaybx b

n n

212.9 2.8 y y a bx

212. 9 2. 840 $324. y


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Recall

Using the formulae from slide 11

i. The correlation coefficient =

i. The coefficient of determination =

Thus 77% of the variation in maintenance costs is explained by the variation in

machine ages.

2 210; 300; 2970; 97650; 12050;

nxy xy x y

2 22 2

2 2

10 97650 300 2970

10 12050 300 10 913050 2970

0.88

n xy x yr

n x x n y y

r

r

22

0.88 0.cdr

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