Question

14. 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).NameTeamAgeHeightGAVEHRSalaryCedric MullinsOrioles25173 cm22.0940$557,500Tim AndersonWhite Sox26185 cm123.33518$1,400,000Christin StewartTigers25183 cm104.23310$556,400Alex GordonRoyals35185 cm150.26613$20,000,000Jonathan SchoopTwins27185 cm121.25623$7,500,000Marcus SemienAthletics29183 cm162.28533$5,900,000Yandy DiazRays28188 cm79.26714$558,400Randal GrichukBlue Jays28188 cm151.23231$5,000,000Josh DonaldsonBraves33185 cm155.25937$23,000,000Joey VottoReds36188 cm142.26115$25,000,000Cody BellingerDodgers24193 cm156.30547$605,000Ryan BraunBrewers35188 cm144.28522$19,000,000Maikel FrancoPhillies27185 cm123.23417$5,200,000Ian KinslerPadres37183 cm87.2179$3,750,000Marcell OzunaCardinals28185 cm130.24129$12,250,000Suppose 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 associationYes, it seems likely that there is a strong negative associationNo, 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).ˆy= 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 accurateThe prediction for 70 home runs is likely to be more accurate

Answer 1

1) Since the question here is to predict a player's salary, then we'll need two columns from that: HR and Salary to begin with:

2) Now we need to expand this table to get the values we need to write out the equation, to begin answering those questions:

a) Examining the data, we can infer that:

No, it does not seem likely that there is a strong correlation

Analyzing Home Runs (x) and Salaries (y) on the table above, since there is one player with 47 HR earning a lot less than another one who made 15 Home Runs.

b) Let's calculate the value of this correlation coefficient using this formula:

`r=\frac{n\Sigma xy-\Sigma x\Sigma y}{\sqrt[]{\lbrack n\Sigma x^2-(\Sigma x)^2\rbrack\lbrack n\Sigma y^2-(\Sigma y)^2\rbrack}}`

Plugging into that the values or our table we have:

`\begin{gathered} r=\frac{n\Sigma xy-\Sigma x\Sigma y}{\sqrt[]{\lbrack n\Sigma x^2-(\Sigma x)^2\rbrack\lbrack n\Sigma y^2-(\Sigma y)^2\rbrack}} \\ r\approx0.137 \end{gathered}`

So we have a correlation (r) of r= 0.137 a weak positive correlation

c)

d) To find the equation of the line, we need to find the slope first and then the linear coefficient:

`\hat{y}=mx+b`

So let's find out the slope, and then plugging into that the values of that table for Σ the last row of each column: Σx, Σy, Σxy, Σx² and Σy² and n=15

`\begin{gathered} m=(n\Sigma xy-\Sigma x\Sigma y)/(n\Sigma x^2-(\Sigma x)^2) \\ m=98257 \end{gathered}`

For the "b" term, i.e. the linear coefficient: