The following data represents the amount of Profit (in thousands of $) made by a trucking company dependent on gas prices. Gas $ Calories Construct and Describe the Scatterplot for this data. 2.Find the Least Square Regression Line

RESIDUALS (also called Errors) The vertical distances from the observed y values and the predicted y on the line are called Residuals. Residual = Observed y – Predicted y Value Residual =

xy Calculating Residuals Find the Residuals for each of the three values

A Residual Plot is a scatter plot where the Residual values are on the y-axis and are plotted against the explanatory variable x (or ). Used to determine if the model (the equation) fits the data. Why do we want to calculate Residuals?????

A Residual Plot that reveals NO Pattern signifies that the model is a GOOD FIT. A Residual Plot that reveals any Pattern signifies that the model should NOT be used for the data. Random scattered points. The Linear Equation is a Good Model. A Pattern is Observed. The Linear Equation is NOT a good Model. Increasing Pattern. The line is NOT a good Model for larger values of x.

Fast food is often considered unhealthy because of the amount of fat and calories in it. Does the amount of Fat content contribute to the number of calories a food product contains? Fat (g) Calories Interpret the residual plot.

Calculating Residuals Fat (g) Calories xyxy Residual plot: The scatter plot depicts a random scatter of points = the model FITS the data well Residual plot:

Fat (g) Calories Residual plot: The scatter plot depicts a random scatter of points = the model FITS the data well Residual plot:

Finding the Least Squares Regression Model from summary statistics :

Here are the summary statistics for the number of hurricanes that have formed per year over the past 100 years and for Temperature of the ocean each year for the same time period. We would like to use ocean temperature to predict number of hurricanes. a.) Find the Least squares regression line AND interpret a and b in contexts to the problem.. means.d. Temperature of the Ocean (degrees Fahrenheit) r = 0.94 Number of Hurricanes 116.2

Homework: Page 192: 1, 38, 42, 43

xy Calculating Residuals

Does adding a fuel additive help gasoline mileage in automobiles? Use Linear Regression to analyze the following data: Amount of STP fuel additive added to the gas tank (in ounces) = x Recorded gas mileage = y 1. Graph the Scatterplot Describe the direction, form and strength. 2. Find the Residuals for all 10 points. Plot them & interpret them. 3. Find the Least Squares Regression Line. In contents to the problem, interpret the meaning of ‘y-int’ and ‘slope’. 4. What is the regression line Correlation Coefficient value? What does this value indicate in relation to the data? 5. What is the Coefficient of Determination value? What does this value indicate in relation to the data? 6. Predict the gas mileage after adding 15 ounces of fuel additive. 7. Find the predicted gas mileage after adding 100 ounces. X Y

R 2 : 91.8% of the variation in the predicted gas milage is attributed by the amount of additive added to the tank. Regression model: Gas Milage = (Add.) Slope: For every additional ounce of Additive the model predicts an additional miles per gallon. y-intercept: If the tank contained NO Gas Additive the car would still get mpg. Correlation: r = Very strong positive linear relationship. R2:R2: Regression model: Slope: y-intercept: Correlation: Residual plot: The scatter plot depicts a random scatter of points = the model FITS the data well Residual plot:

WARM-UP Is there an association between how much a baseball team pays its players (Average in millions) and the team winning percentage? Find AND interpret r and R 2. Team Average Win PCT N.Y. Yankees Boston Texas Arizona Los Angeles New York Mets Atlanta Seattle Cleveland San Francisco Toronto Chicago Cubs St. Louis St. Louis Examine Graph ŷ = x r = 0.31 R 2 = 9.8%

R-Sq. = = 9.8% 9.8% of the variation in the predicted values of Winning Percent is attributed to the salaries of the players. 90.2% of that variation is attributed by other factors.

1. Graph the Scatterplot Describe the direction, form and strength. 2. Find the Residuals for all 10 points. Plot them & interpret them. 3. Find the Least Squares Regression Line. In contents to the problem, interpret the meaning of ‘a’ and ‘b’. 4. What is the regression line Correlation Coefficient value? What does this value indicate in relation to the data? 5. What is the Coefficient of Determination value? What does this value indicate in relation to the data? 6.Predict the gas mileage after adding 15 ounces of fuel additive. 7.Find the predicted gas mileage after adding 100 ounces. This is called EXTRAPOLATION when you make predictions for data outside your range.