Question

Predicting wins Earlier we investigated the relationship between x =

payroll (in millions of dollars) and y = number of wins for Major League Baseball teams in 2016. Here is a scatterplot of the data, along with the regression line y^=60.7+0.139x

a. Predict the number of wins for a team that spends \(200 million on payroll.

b. Predict the number of wins for a team that spends \)400 million on payroll.

c. How confident are you in each of these predictions? Explain your reasoning.

Short answer

Part (a) The number of wins is$88.5$

Part (b) The number of wins is$116.3$

Part (c) we need to check how confident are we in the results of parts (a) and (b).

Step-by-step solution

Part (a) Step 1: Given information

It is stated in the regression line question that,

$\stackrel{⏜}{y}=60.7+0.139x$

Part (a) Step 2: Concept

The most common method for fitting a line to a scatterplot is least squares.

Part (a) Step 3: Calculation

As a result, the number of victories for a team with a salary of $\$200$million is computed as follows:

$$\begin{gathered} \stackrel{⏜}{y}=60.7+0.139x \\ =60.7+0.139\left(200\right) \\ =88.5 \end{gathered}$$

As a result, for a team with a salary of $\$200$ million, the expected number of victories is $88.5$

Part (b) Step 1: Calculation

The regression line is provided in the question.

$\stackrel{⏜}{y}=60.7+0.139x$

Thus, the number of wins for a team that spends $\$400$million on payroll is calculated as:

$$\begin{gathered} \stackrel{⏜}{y}=60.7+0.139x \\ =60.7+0.139\left(400\right) \\ =116.3 \end{gathered}$$

As a result, for a team with a salary of $\$400$million, the expected number of victories is $116.3$

Part (c) Step 1: Explanation

The regression line is provided in the question,

$\stackrel{⏜}{y}=60.7+0.139x$

As a result, we must assess our confidence in the results of parts (a) and (b) (b). We can see from the scatterplot that the payroll in the data set appears to range from $\$75$million to $275$million dollars.

We are confident in the forecast of the portion since $200$million dollars is within the I range of payrolls in the data set (a). We are not confident in the prediction of part (b) since we are using extrapolation and $\$400$million is not within the range of payrolls in the data set.