91Ó°ÊÓ

In baseball, is there a linear correlation between batting average and home run percentage? Let \(x\) represent the batting average of a professional baseball player, and let \(y\) represent the player's home run percentage (number of home runs per 100 times at bat). A random sample of \(n=7\) professional baseball players gave the following information (Reference: The Baseball Encyclopedia, Macmillan Publishing Company). $$ \begin{array}{l|lllllll} \hline x & 0.243 & 0.259 & 0.286 & 0.263 & 0.268 & 0.339 & 0.299 \\ \hline y & 1.4 & 3.6 & 5.5 & 3.8 & 3.5 & 7.3 & 5.0 \\ \hline \end{array} $$ (a) Make a scatter diagram and draw the line you think best fits the data. (b) Would you say the correlation is low, moderate, or high? positive or negative? (c) Use a calculator to verify that \(\Sigma x=1.957, \Sigma x^{2} \approx 0.553, \Sigma y=30.1\), \(\Sigma y^{2}=150.15\), and \(\Sigma x y \approx 8.753 .\) Compute \(r .\) As \(x\) increases, does the value of \(r\) imply that \(y\) should tend to increase or decrease? Explain.

Step by step solution

01

Create a Scatter Plot

Plot the given pairs of batting average (x) values against home run percentage (y) on a graph. Place the batting average on the horizontal axis and the home run percentage on the vertical axis. Look for a general trend in the plotted points.

02

Fit a Line to the Data Points

Visually assess the plotted points and draw a straight line that best represents the relationship between the batting averages and home run percentages. The line should have an approximately equal number of points on either side.

03

Description of Correlation

Examine the orientation and closeness of the data points to the line. If points are tightly clustered around the line, the correlation is high. If they are more scattered, it is moderate or low. Also indicate the direction of the relationship: positive (upward slope) or negative (downward slope).

04

Calculate the Correlation Coefficient r

Use the formula for the Pearson correlation coefficient: \[ r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n \Sigma x^2 - (\Sigma x)^2][n \Sigma y^2 - (\Sigma y)^2]}} \]Substitute the given values: \( n=7, \Sigma x=1.957, \Sigma x^2 \approx 0.553, \Sigma y=30.1, \Sigma y^2=150.15, \Sigma xy\approx 8.753 \).Calculate \(r\) to determine the strength and direction of the correlation.

05

Interpret the Correlation Coefficient

Analyze the computed value of \(r\). If \(0.7 \leq |r| \leq 1\), then the correlation is strong; if \(0.4 \leq |r| < 0.7\), it is moderate; and if \(|r| < 0.4\), it is weak. A positive \(r\) value indicates that as \(x\) increases, \(y\) tends to increase as well. A negative \(r\) implies \(y\) decreases as \(x\) increases.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatter Plot

A scatter plot is an essential tool in statistics when you want to visually examine the relationship between two variables. In this case, it's used to explore the connection between batting average (x) and home run percentage (y) for baseball players.
To create a scatter plot, you start by plotting each pair of values. Here, the batting averages are placed on the horizontal axis and the home run percentages on the vertical axis. Each data point is represented by a dot on the graph. This visual representation can quickly show you if there is a pattern or trend.
Usually, the points on a scatter plot may form a pattern that suggests a linear or non-linear relationship. For this exercise, if the plotted points approximate a straight line, it might indicate a linear relationship between batting averages and home run percentages, which can be further explored with statistical tests for correlation.

Pearson Correlation Coefficient

The Pearson correlation coefficient, denoted as \(r\), is a statistical measure that describes the strength and direction of the relationship between two variables.
The formula for its calculation in this context is:

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

By substituting the calculated values \(n=7, \Sigma x=1.957, \Sigma x^2 \approx 0.553, \Sigma y=30.1, \Sigma y^2=150.15, \) and \(\Sigma xy\approx 8.753\), \(r\) can be computed.
The value of \(r\) lies between -1 and +1. A positive \(r\) value suggests that as one variable increases, the other tends to increase as well, indicating a positive correlation. Conversely, a negative \(r\) value implies that as one variable increases, the other tends to decrease, showing a negative correlation. It's a precise tool for quantifying how well two variables are related.

Linear Regression

Linear regression is a method used to identify and model the linear relationship between two variables. It allows the prediction of one variable based on the value of another.
In the context of this exercise, linear regression would involve deriving an equation for a straight line that best fits the scatter plot data.
The general form of a linear equation is:

• \[ y = mx + c \]

where \(m\) is the slope of the line, representing the change in home run percentage for a unit change in batting average. The \(c\) is the y-intercept and shows the value of \(y\) when \(x=0\).
By computing this regression line using the exercise data, you can estimate the expected home run percentage given a specific batting average. It's a valuable analysis when predicting future outcomes based on current data.

Batting Average Analysis

Batting average is a key performance indicator in baseball, reflecting a player's hitting ability. It is calculated as the number of hits divided by the number of times at bat. In this exercise, it serves as one side of the analysis to determine its relationship with another metric: home run percentage.
Analyzing batting average alongside home run percentage can give insights into a player's overall strength and batting style. A higher batting average might suggest a consistent hitter, while an increase in home run percentage could indicate power to hit the ball well.
Correlation and regression analyses are crucial here as they reveal if changes in a player's batting average are associated with changes in their home run percentage. Understanding this relationship can help coaches and analysts make more informed decisions about player development strategies and game tactics.