Confidence and Prediction interval estimates
5. Condence and prediction interval estimates You are a starting pitcher in the major leagues. It's January 2008, and you are in the process of negotiating your salary for the 2008 season. You hire a statistician to help you with your negotiations. She species the following simple linear regression model: Y : n + 1x + E, where y : 2008 salary (in millions of dollars), and x : performance during the regular 2007 season. Then she selects a random sample of 50 major league starting pitchers with signed salary agreements for the 2008 season. She collects data on their 2007 performances and their 2008 salaries and estimates her model several times, each time with a different performance variable. The performance variable that provides the highest R2 is innings pitched (1P). You are not surprised at this result, because in major league baseball, the better a starting pitcher performs, the longer he remains in the game. The sample data for 2008 salary and 2007 IP are located in the Pitchers data set shown in the following DataView tool. [1P data source: Canadian Business Online; Salary data source: USATODAY.com.] Sample Variables = 1 Observations = 50 Sample of Major League Starting Pitchers Data Sources: Canadian Business Online (Innings Pitched) and USATODAY. com (Salary) > i Observations Variable V Type 7 Form V Values 7 Missing V 2008 Salary Quantitative Numeric 50 D Innings Pitched in 2007 Quantitative Numeric 50 D Variable Variable Correlation Correlation On the Data Set panel in the tool, click on the Observations button to view the Pitchers data set. Then click on one of the Correlation panels. Select 2008 Salary in the dropdown menu for Dependent Variable, and select Innings Pitched in 2007 in the dropdown menu for Independent Variable. Click on the Scatter button to view the scatter diagram. Mouse over the least-squares line (in red) to obtain its equation from the label. Click on the Linear Regression button to view the regression statistics. The estimated regression equation is also shown on this page. You pitched 200 innings during the 2007 regular season. During your salary negotiations, you tell the team's owners that the mean 2008 salary of all starting pitchers who pitched 200 innings in the 2007 regular season is unknown, but its point estimate is You are given that S, = 4.51685. When Xg = 200, y = bo + bix has a mean equal to the unknown E(y) and an estimated standard deviation equal to V . [(Hint: Use the tool to find x, the mean number of innings pitched by the pitchers in the sample, and the standard deviation of innings pitched. Square the standard deviation to get the variance. )] Use the Distributions tool to help you answer some of the questions that follow. t Distribution Degrees of Freedom = 26 O O You go on to tell the team owners that they can be 95% confident that the mean salary for all starting pitchers who pitched 200 innings last season is between and Suppose that instead of estimating the mean 2008 salary of all major league starting pitchers who pitched 200 innings in 2007, you predict the 2008 salary of an individual starting pitcher who pitched 200 innings in 2007. The estimated standard deviation corresponding to the prediction of the 2008 salary for one specific pitcher who pitched 200 innings in 2007 is You demand a salary halfway between the midpoint and the upper limit of the 95% prediction interval for the starting salary of an individual starting pitcher who pitched 200 innings. Your salary demand isYour team's owners, who have hired their own statistician to advise them, respond by saying that the regression results are too imprecise to base your pay on the prediction interval. Their counteroffer is to pay you a base salary plus an add-on. Your base salary is equal to the point estimate of the mean salary for starting pitchers who pitched 200 innings last season. To compute your add-on, they'll let xg equal the value of x that provides the most precise condence and prediction interval estimates. Then they'll calculate the bound on the error of estimation associated with the 95% condence interval for E(y). Your add-on is half the bound on the error of estimation. (Hint: Consider the value of X3 that will set (xg if! / 2(xi i) 2 to zero.) The counteroffer made by your team's owners is a salary of V
