Question

5. The table below describes a sample of 15 players in Major League Baseball, chosen from the starting lineups of teams in 2019. The table shows the team, age, position, height, and salary for each player, as well as several statistics from that season. These include the number of games they played (G), their batting average (AVE) (the proportion of their at-bats for which they got a hit), and their home runs (HR). ? Name Team Age Height G AVE HR Salary Cedric Mullins Orioles 25 173 cm 22 .094 0 $557,500 Tim Anderson White Sox 26 185 cm 123 .335 18 $1,400,000 Christin Stewart Tigers 25 183 cm 104 .233 10 $556,400 Alex Gordon Royals 35 185 cm 150 .266 13 $20,000,000 Jonathan Schoop Twins 27 185 cm 121 .256 23 $7,500,000 Marcus Semien Athletics 29 183 cm 162 .285 33 $5,900,000 Yandy Diaz Rays 28 188 cm 79 .267 14 $558,400 Randal Grichuk Blue Jays 28 188 cm 151 .232 31 $5,000,000 Josh Donaldson Braves 33 185 cm 155 .259 37 $23,000,000 Joey Votto Reds 36 188 cm 142 .261 15 $25,000,000 Cody Bellinger Dodgers 24 193 cm 156 .305 47 $605,000 Ryan Braun Brewers 35 188 cm 144 .285 22 $19,000,000 Maikel Franco Phillies 27 185 cm 123 .234 17 $5,200,000 Ian Kinsler Padres 37 183 cm 87 .217 9 $3,750,000 Marcell Ozuna Cardinals 28 185 cm 130 .241 29 $12,250,000 ? ? Suppose that we want to try to predict a player's salary based on the number of home runs they hit (HR). ? (a) Before doing any calculations, does it seem likely that there will be a strong association between these two variables? If so, which direction do you expect for the association? ? Yes, it seems likely that there is a strong positive association Yes, it seems likely that there is a strong negative association No, it does not seem likely that there is a strong association ? ? (b) Calculate the value of the correlation coefficient, r , using a calculator. ? r = ? ? (c) Interpret the value of r : the correlation is Select an answer and Select an answer . ? (d) Find the equation of the regression line for this association (note: this may not be meaningful, depending on the value of r , but we can still use it for practice). ? ˆ y = x + ? ? (e) Ignoring the possibility that the regression line may not be a good fit for the data, use this regression line to predict the salary of a player who hits 21 home runs. Then predict the salary of a player who hits 70 home runs. Which prediction is likely to be more accurate? ? Predicted salary for 21 home runs: $ ? ? Predicted salary for 70 home runs: $ ? ? The prediction for 21 home runs is likely to be more accurate The prediction for 70 home runs is likely to be more accurate ?

Answer 1

(a) No, it does not seem likely that there is a strong association.

Taking, for example, the next values:

HR salary

0 557,500

9 3,750,000

10 556,400

13 20,000,000

47 605,000

we can see that salaries vary randomly

(b) The plot of data and the regression line is shown in the next picture:

where x represents home runs (HR) and y represents Salary

r = 0.1369

(c) the correlation is positive (the slope of the regression line is positive) and negligible (r is near zero)

(d) The equation of the regression line is:

`\hat{y}=98257.3\hat{x}+6,602,100`

(e) Substituting with x^ = 21:

`\begin{gathered} \hat{y}=98257.3\cdot21+6,602,100 \\ \hat{y}=8,665,503.3\text{ \$} \end{gathered}`

Substituting with x^ = 70:

`\begin{gathered} \hat{y}=98257.3\cdot70+6,602,100 \\ \hat{y}=13,480,111\text{ \$} \end{gathered}`

The prediction for 21 home runs is likely to be more accurate