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.