91Ó°ÊÓ

Players, teams, and fans are interested in seeing their leading scorers score lots of points, yet at the same time the number of personal fouls they commit tends to limit their playing time. For the leading scorer on each team, the table lists the number of minutes per game, MPG, and the number of personal fouls committed per game, PFPG, during the \(2008 / 2009\) NBA season.$$\begin{array}{lll|lll} \text { Team } & \text { MPG } & \text { PFPG } & \text { Team } & \text { MPG } & \text { PFPG } \\ \hline \text { Hawks } & 39.6 & 2.23 & \text { Bucks } & 36.4 & 1.36 \\ \text { Celics } & 37.5 & 2.65 & \text { Timberwolves } & 36.7 & 2.82 \\ \text { Hornets } & 37.6 & 2.96 & \text { Nets } & 36.1 & 2.38 \\ \text { Bulls } & 36.6 & 2.24 & \text { Hornets } & 38.5 & 2.72 \\ \text { Cavaliers } & 37.7 & 1.72 & \text { Knicks } & 29.8 & 2.78 \\ \text { Mavericks } & 37.7 & 2.17 & \text { Thunder } & 39.0 & 1.81 \\ \text { Nuggels } & 34.5 & 2.95 & \text { Magic } & 35.7 & 3.42 \\ \text { Pistons } & 34.0 & 2.63 & \text { 76ers } & 39.9 & 1.85 \end{array}$$.a. Construct a scatter diagram. b. Describe the pattern displayed. Are there any unusual characteristics displayed? c. Calculate the correlation coefficient. d. Does the value of the correlation coefficient seem reasonable?

Step by step solution

01

Constructing a scatter diagram

The scatter diagram is constructed by plotting the number of minutes per game (MPG) on the x-axis and the number of personal fouls committed per game (PFPG) on the y-axis. Each dot on the diagram represents a team.

02

Observing the scatter diagram

After plotting the data, observe if there is any pattern or upward or downward trend. Also note if there are any outliers or odd groupings of data points.

03

Calculating the correlation coefficient

Correlation coefficient can be calculated using a basic statistical formula: \(R = N(\Sigma xy) – (\Sigma x)(\Sigma y) / sqrt{ [ N(\Sigma x^2) – (\Sigma x)^2 ] [ N(\Sigma y^2) – (\Sigma y)^2] }\). Here, Σxy is the sum of the product of x and y, Σx is the sum of x, Σy is the sum of y, and N is the number of values.

04

Interpreting the correlation coefficient

The value of correlation coefficient lies between -1 and 1, where -1 indicates a perfect negative linear relationship, 1 indicates a perfect positive linear relationship, and 0 indicates no linear relationship. Using this understanding, we now interpret the calculated correlation coefficient: whether it is near -1, 0, or 1 and how reasonable the value is.

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Key Concepts

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

Scatter Diagram

A scatter diagram is a useful visualization tool in statistics. It helps us understand the relationship between two quantitative variables by plotting them on a flat surface. In this exercise, minutes per game (MPG) is plotted on the x-axis and personal fouls per game (PFPG) on the y-axis.
Each dot represents the leading scorer from a different NBA team during the 2008/2009 season. The scatter diagram is the first step in examining if there's any correlation between playing time and fouls committed.
Looking at the scatter diagram, we try to observe any common patterns or trends as they help in forming initial hypotheses about the relationship. If the dots show any clear upward or downward trend, it might suggest a linear relationship between MPG and PFPG.

• Dots clustered in a tight formation suggest a strong correlation.
• A more spread out pattern indicates a weaker correlation.
• Outliers are individual points that fall far from the others and can greatly affect the correlation analysis.

Minutes Per Game

Minutes per game (MPG) is a measure of how long a player is on the court, on average, for each game they play. It's an important statistic for understanding a player's contribution to the game.
For top scorers, minutes played can vary significantly, impacting their opportunity to score or commit personal fouls.
In any statistical analysis, it is crucial to consider how minutes per game might correlate with other factors like performance, winning records, or in this case, personal fouls.
This measurement helped us decide what to plot on our scatter diagram's x-axis. As each team's leading scorer's MPG was analyzed, we noticed variations that could hint at differences in coaching strategies or player roles within a team. More minutes often allow for additional scoring chances, but also increase the likelihood of committing more fouls.
Analyzing this data with PFPG helps us identify if more playing time correlates with a higher or lower rate of fouling.

Personal Fouls Per Game

Personal fouls per game (PFPG) are a key statistic to consider when evaluating a player's discipline and defensive strategy on the court. Fouls can limit a player's time in the game, as accumulating too many can result in disqualification.
In our scatter diagram, PFPG is placed on the y-axis, giving us insight into how fouls correlate with playing time. Each dot on the y-axis shows one team’s leading scorer’s fouls per game.
Understanding fouls is critical because:

• They impact both personal performance and game outcomes.
• Frequent fouling can hinder a player's ability to stay on the court longer.
• Coaches may adjust playing minutes to manage foul trouble strategically.

By examining this along with MPG, we can derive insights into whether longer playing times result in more fouls or if players adjust their style to prevent them.

Statistical Analysis

Statistical analysis aids in interpreting data from the scatter diagram, giving us more concrete insights. In this context, the correlation coefficient becomes a crucial aspect.
This numerical value ranges from -1 to 1 and tells us how strongly two variables are related.

• A positive correlation signifies that as one variable increases, the other does too.
• A negative correlation suggests that as one variable increases, the other decreases.
• A correlation near zero implies no linear relationship.

After plotting the scatter diagram, we calculate the correlation coefficient using a standard formula. This mathematical computation offers a more precise understanding of the data than the visual scatter plot alone.
The formula considers sums and products of the x (MPG) and y (PFPG) values along with the number of observations. By evaluating this coefficient, we can conclude if our initial observations from the scatter plot match with what is statistically significant. This is pivotal in deciding if the pattern is just incidental or holds a meaningful insight.