2. Equation for simple linear
regression
What is linear Regression?
The least squares linear regression is a method for predicting
the value of a dependent variable y, based on a value of the
independent variably x.
Equation: y*= a + bx
Where: y*= predicted value of the dependent variable
a= constant
b= slope (regression coefficient)
x= value of independent variable
3. Estimation Requirements
• Dependent variable y has a linear
relationship to the independent variable x.
• For each value of x, the probability
distribution of y has the same standard
deviation
• For any given value of x,
– y values are independent
– y values are approximately normally distributed
4. Properties of a Regression Line
• With the regression parameters defined
before for bo and b1
• The line minimizes the sum of the squared differences between the observed
values )y-values) and the predicted values(y* values computed by the regression
equation)
• The regression line passes through the mean of the x-values (x) and through the
mean of the y values (y)
• Regression constant (b0) is equal to y-intercept of regression line
• Regression coefficient (b1) is the average change in the dependent variable (y) for
a 1-unit change in the independent variable (x). It is the slope of the regression
line
• The least squares regression line is the only straight line that has all of these
properties
5. Standard Error
The measure of the average amount that the
regression equation over or under predicts. The
higher the coefficient of determination, the
lower the standard error and the more accurate
predictions are likely to be.
(this will always be given to you)
Variability of Slope Estimate:
Calculator provides standard error of slope as
regression analysis output
6. How to find the Confidence Interval
for the slope of a regression line
Same approach as previous confidence intervals,
BUT the critical value is based on a t-score
with the degree of freedom as n-2.
1- sample statistic regression slope b,
calculated from sample data
2- confidence level
3- Margin of Error
4- Confidence Interval: sample statistic + ME
written as: (Sample Stat – ME, Sample Stat + ME)
7. How to find the Confidence Interval
for the slope of a regression line
Same approach as previous confidence intervals,
BUT the critical value is based on a t-score
with the degree of freedom as n-2.
1- sample statistic regression slope b,
calculated from sample data
2- confidence level
3- Margin of Error
4- Confidence Interval: sample statistic + ME
written as: (Sample Stat – ME, Sample Stat + ME)
