Question

7. 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 1,400,000 Christin Stewart Tigers 25 183 cm 104 .233 10 20,000,000 Jonathan Schoop Twins 27 185 cm 121 .256 23 5,900,000 Yandy Diaz Rays 28 188 cm 79 .267 14 5,000,000 Josh Donaldson Braves 33 185 cm 155 .259 37 25,000,000 Cody Bellinger Dodgers 24 193 cm 156 .305 47 19,000,000 Maikel Franco Phillies 27 185 cm 123 .234 17 3,750,000 Marcell Ozuna Cardinals 28 185 cm 130 .241 29 $12,250,000 ? ? Suppose that we want to try to predict the number of home runs that a player hits (HR) based on the number of games they play (G). ? (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 (positive or negative) and (strong or moderate or weak or negligible). ? (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 how many home runs a player will hit if he plays 100 games. ?

Answer 1

Step 1: Define a Scatter Plot

To answer this part of the question without calculation, we need a scatterplot of the data

Step 2: (a) Check for association using scatter plott

In our case, the scatter plot is as shown in the image below

As you can see in the image, the points have a linear association.

Hence, the correct answer for (a) is:

Yes, it seems likely that there is a strong positive association

Step 3 (b): Compute the correlation coefficient with a calculator

Using a calculator, the correlation coefficient, r, is given by

`r=0.7603`

r = 0.7603

Step 4 (c): Interprete the value of r.

The value of r can be interpreted using the table below

Size of correlation | Interpretation

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0.9 to 0.1 (-0.9 to -1.0) | Very high positive (negative) correlation

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0.7 to 0.9 (-0.7 to -0.9) | High positive (negative) correlation

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0.5 to 0.7 (-0.5 to -0.7) | Moderate positive (negative) correlation

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0.3 to 0.5 (-0.3 to -0.5) | Low positive (negative) correlation

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0.0 to 0.3 (-0.0 to -0.3) | Negligible correlation

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In our case, r = 0.7603.

Hence, using the table, we can conclude that

the correlation is positive and negiligible

Step 5: (d) Find the equation of the regression line

With a calculator, we can find the equation of regression line as

`\bar{y}=0.249064z-9.50126`

Step 6: (e) Predict how many home runs a player will hit if he plays 100 games

If a player plays 100 games, then

`\begin{gathered} z=100 \\ \text{then} \\ \bar{y}=0.249064(100)-9.50126\approx15 \end{gathered}`

Hence, if a player plays 100 games, he will hit

15 HR (home runs)