The following data represents the amount of Profit (in thousands of $) made by a trucking company dependent on gas prices. Gas $ 1.90 1.95 2.10 2.50 1.80 2.40 2.05 Profit 420 300 220 120 500 180 433 Construct and Describe the Scatterplot for this data. Find the Least Square Regression Line. Interpret the slope and y-intercept. Interpret R2

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 =

Calculating Residuals Find the Residuals for each of the three values. x y 2 50 3 48 5 30 52 -2 45 3 31 -1

Why do we want to calculate Residuals????? 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. Used to determine if the model is appropriate.

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 all the data.

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) 19 31 34 35 39 26 43 Calories 920 1300 1310 960 1180 1100 1260 Interpret the residual plot by using Fat (g) as the independent Var. and RESIDUALS as the Dependent Var.

Calculating Residuals x y Fat (g) 19 31 34 35 39 26 43 Calories 920 1300 1310 960 1180 1100 1260 997.6 1131.2 1164.6 1175.8 1220.3 1075.5 1264.9 -77.6 168.8 145.4 -215.8 -40.3 24.5 -4.9 Residual plot: Residual plot: The scatter plot depicts a random scatter of points = the model FITS the data well

Homework: Page 193: 38, 42, 43

Homework: Page 193: 38, 42, 43

Homework: Page 193: 38, 42, 43

Finding the Least Squares Regression Model from summary statistics :

Temperature of the Ocean (degrees Fahrenheit) 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.. mean s.d. Temperature of the Ocean (degrees Fahrenheit) 62 3.8 r = 0.94 Number of Hurricanes 11 6.2

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

Calculating Residuals x y 50 1 48 2 30 3 40 4 20 5 35 6 10 7 12 49.83 0.17 44.35 3.65 38.86 -8.86 33.37 6.63 27.88 -7.88 22.39 12.61 16.91 -6.91 11.42 0.58

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 4 6 8 10 12 13 16 20 22 24 Y 14.4 15.3 16.1 15.8 17 17.5 17.6 19.1

Regression model: Regression model: Gas Milage = 12.898 + 0.243(Add.) Residual plot: Residual plot: The scatter plot depicts a random scatter of points = the model FITS the data well Slope: Slope: For every additional ounce of Additive the model predicts an additional 0.243 miles per gallon. y-intercept: y-intercept: If the tank contained NO Gas Additive the car would still get 12.898 mpg. Correlation: Correlation: r = 0.958 Very strong positive linear relationship. R2: R2: 91.8% of the variation in the predicted gas milage is attributed by the amount of additive added to the tank.

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 R2 . Team Average Win PCT N.Y. Yankees 4.34 64.0 Boston 3.61 57.4 Texas 3.63 44.4 Arizona 3.11 60.5 Los Angeles 3.64 56.8 New York Mets 3.63 46.6 Atlanta 3.01 63.1 Seattle 3.21 57.4 Cleveland 2.63 45.7 San Francisco 2.89 59.0 Toronto 2.65 48.1 Chicago Cubs 2.70 41.4 St. Louis 2.84 59.9 Examine Graph ŷ = 38.953 + 4.725x r = 0.31 R2 = 9.8%

R-Sq. = 0.098 = 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? 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.