Pitchers The table shows the Earned Run Average (ERA) and...

Pitchers The table shows the Earned Run Average (ERA) and WHIP rating (walks plus hits per inning) for the top 40 Major League Baseball pitchers in the 2017 season. Top pitchers will tend to have low ERA and WHIP ratings. (Source: ESPN.com) a. Make a scatterplot of the data and state the sign of the slope from the scatterplot. Use WHIP to predict ERA. b. Use linear regression to find the equation of the best-fit line. Show the line on the scatterplot using technology or by hand. c. Interpret the slope. d. Interpret the $y$ -intercept or explain why it would be inappropriate to do so. $$ \begin{aligned} &\begin{array}{|ll|} \hline \text { WHIP } & \text { ERA } \\ \hline 0.87 & 2.25 \\ \hline 0.95 & 2.31 \\ \hline 0.9 & 2.51 \\ \hline 1.02 & 2.52 \\ \hline 1.15 & 2.89 \\ \hline 0.97 & 2.9 \\ \hline \end{array}\\ &\begin{array}{|c|c|} \hline \text { WHIP } & \text { ERA } \\ \hline 1.21 & 3.55 \\ \hline 1.22 & 3.64 \\ \hline 1.22 & 3.66 \\ \hline 1.27 & 3.82 \\ \hline 1.15 & 3.83 \\ \hline 1.16 & 3.86 \\ \hline \end{array} \end{aligned} $$

Pitchers The table shows the Earned Run Average (ERA) and WHIP rating (walks plus hits per inning) for the top 40 Major League Baseball pitchers in the 2017 season. Top pitchers will tend to have low ERA and WHIP ratings. (Source: ESPN.com)
a. Make a scatterplot of the data and state the sign of the slope from the scatterplot. Use WHIP to predict ERA.
b. Use linear regression to find the equation of the best-fit line. Show the line on the scatterplot using technology or by hand.
c. Interpret the slope.
d. Interpret the $y$ -intercept or explain why it would be inappropriate to do so.
$$
\begin{aligned}
&\begin{array}{|ll|}
\hline \text { WHIP } & \text { ERA } \\
\hline 0.87 & 2.25 \\
\hline 0.95 & 2.31 \\
\hline 0.9 & 2.51 \\
\hline 1.02 & 2.52 \\
\hline 1.15 & 2.89 \\
\hline 0.97 & 2.9 \\
\hline
\end{array}\\
&\begin{array}{|c|c|}
\hline \text { WHIP } & \text { ERA } \\
\hline 1.21 & 3.55 \\
\hline 1.22 & 3.64 \\
\hline 1.22 & 3.66 \\
\hline 1.27 & 3.82 \\
\hline 1.15 & 3.83 \\
\hline 1.16 & 3.86 \\
\hline
\end{array}
\end{aligned}
$$

Key Concepts

Interpretation of Y-intercept

The y-intercept is the predicted value of the dependent variable when the independent variable is zero. Its interpretation must be contextual; if a value of zero for the independent variable is not realistic or falls outside the observed range, interpreting the y-intercept may not provide meaningful or practical insights.

Interpretation of Slope

The slope in a linear regression equation quantifies the change in the dependent variable for each one-unit change in the independent variable. It provides insight into the strength and direction of the relationship, indicating how much the dependent variable is expected to increase or decrease as the independent variable changes by one unit.

Correlation and Slope Sign

Correlation measures the strength and direction of a linear relationship between two quantitative variables. The sign of the slope indicates whether the relationship is positive or negative; a positive slope suggests that as the independent variable increases, the dependent variable increases, and vice versa for a negative slope.

Scatterplot

A scatterplot is a graphical display that uses dots to represent the values obtained for two different variables, one plotted along the x-axis and the other along the y-axis. This visualization helps in identifying the type of relationship, if any, between the two variables, including trends, patterns, clusters, and outliers.

Linear Regression

Linear regression is a statistical method used for modeling the relationship between a dependent variable and one or more independent variables by fitting a linear equation to the observed data. The best-fit line obtained minimizes the distances between the observed values and the line, typically by minimizing the sum of the squared differences (errors).

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