- You Are A Starting Pitcher In The Major Leagues It S January 2008 And You Are In The Process Of Negotiating Your Salar 1 (52.26 KiB) Viewed 35 times
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 180 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 180 innings in the 2007 regular season is unknown, but its point estimate is You are given that the standard error of the estimate, s = 4.51685. When x* = 180, y* = bo + bıx* has a mean equal to the unknown E[y*) and an estimated standard deviation equal to $y* = . (Hint: Use the tool to find 7, 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, and multiply the variance by (n-1) to find (Xi - x)2.] Use the Distributions tool to help you answer some of the questions that follow. t Distribution Degrees of Freedom = 26 . -3 -2 -1 3 an You go on to tell the team owners that they can be 95% confident that the mean salary for all starting pitchers who pitched 180 innings last season is between and Suppose that instead of estimating the mean 2008 salary of all major league starting pitchers who pitched 180 innings in 2007, you predict the 2008 salary of an individual starting pitcher who pitched 180 innings in 2007. The estimated standard deviation corresponding to the prediction of the 2008 salary for one specific pitcher who pitched 180 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 180 innings. Your salary demand is
You go on to tell the team owners that they can be 95% confident that the mean salary for all starting pitchers who pitched 180 innings last season is between and Suppose that instead of estimating the mean 2008 salary of all major league starting pitchers who pitched 180 innings in 2007, you predict the 2008 salary of an individual starting pitcher who pitched 180 innings in 2007. The estimated standard deviation corresponding to the prediction of the 2008 salary for one specific pitcher who pitched 180 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 180 innings. Your salary demand is Your 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 180 innings last season. To compute your add-on, they'll let x* equal the value of x that provides the most precise confidence and prediction interval estimates. Then they'll calculate the margin of error associated with the 95% confidence interval for E[y*). Your add-on is half the margin of error. (Hint: The value of ŷ is smallest when ** = 1.) . The counteroffer made by your team's owners is a salary of