12. Computing the regression line and making predictions Suppose you are a dolphin trainer at SeaWorld. You teach the dolphins by rewarding them with fish treats after each successful attempt at a new trick. The following table lists the dolphins, the number of treats per success given to each, and the average number of attempts necessary for each to learn A-Z to perform the tricks. Dolphin Number of Treats Number of Attempts Diana 2 6 Frederick 5 Fatima Co W H UT Marlin You can use the preceding sample data to obtain the regression line, where Y is the predicted value of Y: Y = bX + a O One formula for the slope of the regression line is as follows: b= SS, To calculate the slope, first calculate SP and SSx :SP = , and SSx = (Hint: For SP use the computational formula and for SSx use the definitional formula.) A-Z The slope of the regression line is , and the Y intercept of the regression line is The difference between Y and Y for a particular sample point (observation) is called a residual. Calculate the predicted Y (Y) for each of the dolphins, and then calculate the residuals. Dolphin Number of Treats Number of Attempts Predicted Y (Y) Residual Diana 2 6 Frederick A UT Fatima CO Marlin W unT On the following scatter diagram of the blue sample points (circle symbol), use the orange points (square symbol) to plot the regression line. Make sure that the orange line spans the entire graph (from left to right). A line segment will automatically connect the points. O 10 Co O Regression Line 7On the following scatter diagram of the blue sample points (circle symbol), use the orange points (square symbol) to plot the regression line. Make sure that the orange line spans the entire graph (from left to right). A line segment will automatically connect the points. A-Z 10 -O- O Regression Line CO O O NUMBER OF ATTEMPTS O N 0 2 3 O NUMBER OF TREATS Use the regression line to estimate the number of trials it would take to learn these tricks if a dolphin received five treats per trick. Y for X = 5 would beThe head dolphin trainer is pressuring you to teach the dolphins many new tricks quickly. She has asked you to use the least-squares regression line to predict how fast the dolphins can learn tricks if you were to give them 9 treats. Which of the following is the most appropriate response? 0 The regression line predicts that the dolphins will need -D.5 attempts to learn a trick if they are given 9 treats O The regression line was estimated using 1 to 4 treats and should not be used to predict what would happen if the dolphins were given 9 treats. O The regression line predicts that the dolphins will need 5.5 attempts to learn a trick if they are given 9 tream. 0 Since the regression line predicts that the dolphins will need a negative number or attempts, you can assume the dolphins need 0 attempts if given 9 treats. Continue without sawing