HW# : Complete the last slide Aim: What are the types of variations of regression models and how do we find them? HW# : Complete the last slide

Do Now: Consider the following hypothetical regression model: x 1 2 3 4 5 y 10 8 12 16 20 What is the equation of the regression line?

Recap: Correlation Coefficient measures the strength of the linear relationship between two variables Gives a degree of association between two or more variables (NOT A CAUSAL RELATIONSHIP) Line of Regression is the line of best fit in a set of data

Types of Variations for the Regression Model From Do Now: The equation of the regression line is y’ = 4.8 + 2.8x For each x, there is an observed y value and a predicted y’ Meaning, according to the chart, when x =1, y = 10 ALTHOUGH is you substitute 1 for x in your equation, y’ = 7.6 y’ = 4.8 + 2.8(1) = 7.6 What does this mean?

What does this mean? Variations? The closer the observed values (from the data) are to the predictive values (y’), the better the fit is and the closer r is to +1 or -1 The difference between the observed values and the predictive values show variations in data

What is total variance? Total Variance: the sum of the squares of the vertical distances each point is from the mean Two types: 1. that which is attributed to the relationship of x and y (explained variance) 2. that which is due to chance (unexplained variance)

Types of variation for the regression model: Explained Variation Explained Variation: the variation that is obtained from the relationship of x and y (from the predictive y’ values) variations that can be explained by the relationship Closer r is to +1 or -1, the better the points fit the line and the closer explained variance is to total variance If all the points fit on the regression line, the explained variance will equal the total variance because y’ would equal to y in each case

Types of variation for the regression model: Unexplained Variation Unexplained Variation: variation due to chance This variation cannot be attributed to the relationship When unexplained variance is small, the value of r is close to +1 or -1. If all points fit on regression line, the unexplained variation will equal 0

THEREFORE… The total variation is equal to the sum of the explained variation and the unexplained variation

What does this all mean?

How do we find the types of variations? From Do Now: x 1 2 3 4 5 y 10 8 12 16 20 Step 1: Find predicted y’ values for EACH of the given observed data

How do we find the types of variations? Step 2: Find the mean of the y values Step 3: Find the total variance

How do we find the types of variations? Step 4: Find the explained variation Step 5: Find the unexplained variation

How do we find the types of variations? Step 6: Find Total Variance

How do we see this on a scatterplot?

Homework: complete this slide A researcher collects the following data and determines the there is a significant relationship between the age of a copy machine and its monthly maintenance cost. The regression equation is y’ = 55.57 + 8.13x. Find the total variation. X Y 1 62 2 78 3 70 4 90 93 6 103