PREPARED BY:PAOLO LORENZO BAUTISTA

Simple Linear Regression and Correlation

SLR : PLBautista

Simple Linear Regression

In SLR, we assume one dependent and one independent variable.

We try to predict the outcome/result of a dependent variable based on the independent variable

Example: The intelligence test scores of 12 college freshmen were obtained, and we try to find out if this has an effect on the freshmen’s chemistry grade.ITS 65 50 55 6

555 70 65 70 55 70 50 55

Chem Grade

85 74 76 90 85 87 94 98 81 91 76 74

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Simple Linear Regression

Intelligence Test Scores and Freshmen Chemistry Grades

0

20

40

60

80

100

120

0 10 20 30 40 50 60 70 80

Intelligence Test Score

Ch

emis

try

Gra

de

Series1

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Simple Linear Regression

We wish to find a line of the formy = a + bx

which will tell us the trend shown by the data.

Using PHStat, we havey = 30.0433 +0.8972x

Predict a freshman’s chemistry grade if his intelligence test score is 70.

What should a freshman’s intelligence test score be if he wants a grade of 90 for chemistry?

Provide an interpretation of the slope.

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Correlation Analysis

Measures the strength of relationships between two variables by means of a number called the correlation coefficient.

Also called the Pearson correlation coefficient.

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Correlation Analysis

Photo from wikipedia

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Correlation Analysis

Example: Find the correlation coefficient from our previous example.

r = 0.8625 indicating a strong positive linear relationship

Coefficient of Determination (r2) – the percentage of the variation in the dependent variable (Y) that can be explained by a linear relationship with the independent variable (X)

r2 = 0.7438 meaning 74.38% of the variation in chemistry grades is explained by a linear relationship with intelligence scores

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Correlation Analysis

May require careful personal inspection

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Hypothesis Testing for Correlation

H0: ρ=0H1: ρ≠0 (significant linear association

between X and Y)t-test with n-2 degrees of freedomTest statistic:

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Exercise

The following data show the amount of money 8 companies used for advertising, and the corresponding sales:

Advertising ($) Sales ($)

40 385

20 400

25 395

20 365

30 475

50 440

40 490

20 420

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Exercise

Provide the estimated regression equation to predict sales given the amount spent on advertising.

Provide an interpretation of the slope.What is the expected sales of a company if it

spends $45 for advertising.Compute the correlation coefficient.

Interpret.Compute the coefficient of determination.

Interpret.Test if there is a significant linear association

between advertising and sales.