1. SIMPLE LINEAR REGRESSION MODEL
CHAPTER 5 :
SIMPLE
LINEAR REGRESSION
MODEL

2. CHAPTER OUTLINE: INTRODUCTION SCATTER PLOT METHOD OF LEAST SQUARES
CORRELATION OF COEFFICIENT

3. 5.3.1 INTRODUCTION:
Regression: is a statistical procedure for establishing the relationship between 2 or more variables.
Linear regression: study on the linear relationship between two or more variables.
2 types of relationship: 1) Simple ( 2 variables)
2) Multiple (more than 2 variables)
Simple linear regression (SLR) model: is a statistical method used to describe the nature of the relationship between two variables.
SLR: One independent variable as opposed to multiple linear regression, which handles two or more independent variables.

4. Commonly used for analyzing mean of response variable, y which changes according to the magnitude of an independent variable, x.
In simple linear regression only two variables are involved:
X is the independent variable.
Y is dependent variable.
Independent / Explanatory variable:
Variable in regression that can be manipulated.
Dependent / Response variable:
Variable in regression that cannot be manipulated.

5. Linear relationship can be positive or negative.
Positive linear relationship: both variables are either increase or decrease at the same time.
Negative linear relationship: as one variable increase, the other variable decrease, and vice verse.
Regression model can be used to see the trend and make prediction of values for the data.
Can be used to evaluate the magnitude change in one variable due to a change in another variable.

6. Example 1:
1) A nutritionist studying weight loss programs might wants to find out if reducing intake of carbohydrate can help a person reduce weight.
a) X is the carbohydrate intake (independent variable).
b) Y is the weight (dependent variable).
2) An entrepreneur might want to know whether increasing the cost of packaging his new product will have an effect on the sales volume.
a) X is cost
b) Y is sales volume

7. 5.3.2 SCATTER PLOTS A scatter plot is a graph or ordered pairs (x,y).
The purpose of scatter plot – to describe the nature of the relationships between independent variable, X and dependent variable, Y in visual way.
The independent variable, x is plotted on the horizontal axis and the dependent variable, y is plotted on the vertical axis.
Scatter plot: graphical way to check either the relationship between the two considered variables exist or not.

8. SCATTER PLOT
Scatter plots are especially useful when there are a large number of data points.
They provide the following information about the relationship between two variables:
(1) Strength
(2) Shape - linear, curved, etc.
(3) Direction - positive or negative
(4) Presence of outliers

9. Examples:

10. Example 2:
Absences and Final Grades Construct and interpret the scatter plot for the data obtained in a study on the number of absences and the final grades of seven randomly selected students from a statistics class. The data are shown here.
Number of absences, x 2 5 6 8 9 12 15
Final grade, y 86 90 82 78 74 58 43

11. The pattern of the points on the scatter plot shows a negative linear relationship between number of absences and final grade. As the number of absences increase, the values of the final grade decrease.

12. 5.3.3 SIMPLE LINEAR REGRESSION
A line of best fit is needed to predict the value of y from the values of x.
The closer the points to the line, the better the fit and the prediction.
Regression line can be used to determine the trend and making predictions.
The line of best fit: The sum of the squares of the vertical distances from each point to the line is at a minimum.

13. 5.3.3 SIMPLE LINEAR REGRESSION
To find the possibility of relationship between two variables: 1) positive linear relationship
2) negative linear relationship
3) no apparent relationship.
When r is positive, the line slopes upward to the right. When r is negative, the line slopes downward to the right.

14. LEAST SQUARES METHOD
Given a random sample of observation, the regression line is given by :
Where:
is an intercept
is the regression coefficient
is the value of the independent variable
is the predicted value of dependent variable.

15. LEAST SQUARES METHOD
Given any value of x, the predicted value of the dependent variable , can be found by substituting x into the equation:

16. LEAST SQUARES METHOD and are the mean of x and y respectively.
i ) y-intercept for the Estimated Regression Equation:
and are the mean of x and y respectively.
where
ii) Slope for the Estimated Regression Equation,

17. LEAST SQUARES METHOD
where

18. Example 3
Find the regression line for the data in Example 2, and sketch the line on the scatter plot of the data.

21. 5.3.4 CORRELATION COEFFICIENT
Also known as Pearson’s product moment coefficient of correlation.
Does not provide any information about the size of the change in one variable.
The symbol for sample correlation coefficient is r, while population correlation coefficient is p.
No units with r and the value of r will remain unchanged if x and y values are switched.
The correlation coefficient computed from the sample data measures the strength and direction of a linear relationship between two quantitative variables.

22. CORRELATION COEFFICIENT
Formula :
where;

23. Properties of :
Values of r is 1/-1 implies perfectly positive / negative linear relationship between x and y.
Values of r close to 1 (r ≥ 0.5) / -1 (r ≤ -0.5) implies there is a strong positive / negative linear relationship between x and y.
Values of r close to 0 implies weak linear relationship between x and y.
Values of r is 0 implies there is no linear relationship between x and y.

24. Examples:

25. Example 4:
Absences and Final Grades Construct and interpret the scatter plot for the data obtained in a study on the number of absences and the final grades of seven randomly selected students from a statistics class. The data are shown here.
Number of absences, x 2 5 6 8 9 12 15
Final grade, y 86 90 82 78 74 58 43

26. Example 4
Refer Example 2, calculate the value of r and interpret its meaning:

28. Solution:
Thus, there is a strong negative linear relationship between number of absences (x) and final grade (y).