Question

The following data represents the winning percentage (the number of wins out of 162 games in a season) as well as the teams Earned Run Average, or ERA.

The ERA is a pitching statistic. The lower the ERA, the less runs an opponent will score per game. Smaller ERA's reflect (i) a good pitching staff and (ii) a good team defense. You are to investigate the relationship between a team's winning percentage - YY, and its Earned Run Average (ERA) - XX.

Winning Proportion - Y	Earned Run Average (ERA) - X
0.623457	3.13
0.512346	3.97
0.635802	3.68
0.604938	3.92
0.518519	4.00
0.580247	4.12
0.413580	4.29
0.407407	4.62
0.462963	3.89
0.450617	5.20
0.487654	4.36
0.456790	4.91
0.574047	3.75

(a) Using R-Studio, create a scatter-plot of the data. What can you conclude from this scatter-plot?

A. There is a negative linear relationship between a teams winning percentage and its ERA.
B. There is a positive linear relationship between a teams winning percentage and its ERA.
C. There is not a linear relationship between the a teams winning percentage and its ERA.


(b) Use R-Studio to find the least squares estimate of the linear model that expressed a teams winning percentage as a linear function of is ERA. Use four decimals in each of your answers.

YˆiY^i =

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  ? + -  

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XiXi

(c) Find the value of the coefficient of determination, then complete its interpretation.

r2=r2=

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(use four decimals)

The percentage of  ? variation standard deviation the mean  in  ? a teams winning percentage a teams earned run average  that is explained by its linear relationship with  ? the teams winning percentage the teams earned run average  is

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%.

(d) Interpret the meaning of the slope term in the estimate of the linear model, in the context of the data.

As a teams  ? winning percentage earned run average  increases by  ? one percentage point one earned run  the teams  ? winning percentage earned run average  will  ? will increase by an average of will decrease by an average of will increase by will decrease by  

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. (use four decimals)

(e) A certain professional baseball team had an earned run average of 3.45 this past season. How many games out of 162 would you expect this team to win? Use two decimals in your answer.

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games won

(f) The team mentioned in part (e) won 91 out of 162 games. Find the residual, using two decimals in your answer.

Answer 1

Solution

> Winning_Proportion=c(0.623457,0.512346,0.635802,0.604938,0.518519,0.580247,0.41358,0.407407,0.462963,0.450617,0.487654,0.45679,0.574047)
> Earned_Run_Average=c(3.13,3.97,3.68,3.92,4,4.12,4.29,4.62,3.89,5.2,4.36,4.91,3.75)

#plot scatter plot
> Scatter_plot=plot(Earned_Run_Average,Winning_Proportion)

A. There is a negative linear relationship between a teams winning percentage and its ERA

#b

> model=lm(Winning_Proportion~Earned_Run_Average)
> model

Call:
lm(formula = Winning_Proportion ~ Earned_Run_Average)

Coefficients:
(Intercept) Earned_Run_Average
0.9621 -0.1073

# Winning_Proportion = 0.9621 - 0.1073 Earned_Run_Average

(e) A certain professional baseball team had an earned run average of 3.45 this past season. How many games out of 162 would you expect this team to win?

#c

> summarymodel=summary(model)
> summarymodel$r.squared
[1] 0.5472422

The percentage of  variation standard deviation the mean  in  a teams winning percentage a teams earned run average  that is explained by its linear relationship with   the teams winning percentage the teams earned run average  is 54.72%.