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. What is the regression line Correlation Coefficient value? What does this value indicate in relation to the data? 3. What is the Coefficient of Determination value? What does this value indicate in relation to the data? 4. Find the Least Squares Regression Line. In contents to the problem, interpret the meaning of ‘a’ and ‘b’. 5. Predict the gas mileage after adding 15 ounces of fuel additive. 6. Find the Residual for x = 10 ounces of STP. 7. Find the predicted gas mileage after adding 100 ounces. This is called EXTRAPOLATION when you make predictions for data outside your range. X Y

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

The Difference between Outliers and Influential Points An Outlier is an observation that lies outside the overall pattern of the other observations. An Influential point is an observation that if removing it would markedly change the position of the regression line. Outlier but NOT Influential Outlier and Influential Correlation and Least Squares Regression line are Nonresistant meaning that they are strongly influenced by extreme outliers that are influential.

RESIDUALS The Least-Squares regression line is a line in which the vertical distances from the observed y values and the predicted y value are as small as possible. These vertical distances are called Residuals. Residual = Observed y – Predicted y Value Residual = Predicted = Observed = x y

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%

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 x y y Residual

Sum of Squares about the Mean - (SSM) If there was absolutely no association between x and y then the regression line would be a horizontal line passing through the mean of y. The variation between the actual data y values and this horizontal line is measured by SSM. Sum of Squares for Error - (SSE) If there is an association between x and y then the regression line would be modeled by y-hat. The variation between the actual data y values and the predicted y values is measured by SSE. R-Squared – the percent or fraction of variation in the values of y that is explained by the least squares regression of y on x. Measure of goodness of fit.

Warm-Up Does temperature effect ice cream sales. A local ice cream shop sold the following amounts on various days: Temperature # of Sales: Construct and Describe the Form and Direction of the Scatterplot. 2.Identify any Outliers. 3.Find and Interpret the correlation.

WARM - UP Many Blogs are declaring that the Government is manipulating Gas Reserves therefore manipulating Gas Prices. To investigate this analyze the following Gas Prices$ President Bush’s Approval Rating %