Question

Attendance 2006. American League baseball games are played under the designated hitter rule, meaning that pitchers, often weak hitters, do not come to bat. Baseball owners believe that the designated hitter rule means more runs scored, which in turn means higher attendance. Is there evidence that more fans attend games if the teams score more runs? Data collected from American League games during the 2006 season indicate a correlation of \(0.667\) between runs scored and the number of people at the game. (http: //mlb.mlb.com) a) Does the scatterplot indicate that it's appropriate to calculate a correlation? Explain. b) Describe the association between attendance and runs scored. c) Does this association prove that the owners are right that more fans will come to games if the teams score more runs?

Short answer

Scatterplot should show a linear trend; correlation is 0.667, indicating a positive association, but does not prove causation for increased attendance.

Step-by-step solution

Assessing the Scatterplot Appropriateness

To determine if it's appropriate to calculate a correlation using the scatterplot, we should check for linearity. The relationship between runs scored and attendance should exhibit a linear pattern. A scatterplot was likely provided or constructed from the data; if it shows a roughly linear trend, then calculating the correlation is appropriate. If the scatterplot does not show a clear linear pattern or if there are outliers, it may not be appropriate to calculate or rely on the correlation alone.

Describing the Association

The given correlation of 0.667 indicates a moderately strong positive association between runs scored and attendance. This means that, generally, as the number of runs scored increases, the attendance at games tends to increase as well. However, correlation does not measure causation or take into account any lurking variables that might be influencing both runs scored and attendance.

Evaluating the Owners' Claim

Even though there is a positive correlation, it does not prove causation. This means we cannot conclusively say that more runs directly cause more fans to attend games. Other factors such as team popularity, promotional events, weather, or ticket prices could also affect attendance. Thus, while the data shows an association, it does not prove the owners' claim that scoring more runs will lead to increased attendance.

Key concepts

Scatterplot

A scatterplot is a powerful tool that helps us visualize the relationship between two numerical variables. In the context of the baseball games, it provides a visual representation of the runs scored against the attendance at the games. Each point on the scatterplot corresponds to a game, with the x-axis representing runs scored and the y-axis representing attendance.
To decide if it's appropriate to calculate a correlation coefficient from a scatterplot, we first examine the pattern of the points:

• Linear Pattern: If the points on the scatterplot appear to follow a straight line, either upward or downward, it suggests a linear relationship, which is necessary for correlation analysis.
• Outliers: Anomalies or outliers might affect the reliability of the correlation. Removing these can give a more accurate representation.

In this case, if the scatterplot indicates a roughly linear trend, it would be appropriate to measure the correlation between runs scored and attendance.

Linear Relationship

A linear relationship implies that as one variable increases, the other tends to increase or decrease at a consistent rate. In the baseball attendance scenario, the correlation coefficient of 0.667 suggests a positive linear relationship between runs scored and attendance.
This means that generally, when more runs are scored, attendance increases, and vice versa.

• Positive Correlation: Indicates that the variables move in the same direction. More runs lead to more fans.
• Strength of Correlation: The value 0.667 marks a moderately strong relationship. Closer to 1 would mean a stronger linear relationship.

While there is a correlation, it's important to remember that correlation itself doesn't tell us why runs and attendance are related—just that they tend to grow together.

Causation vs. Correlation

One of the most important distinctions in statistical analysis is understanding the difference between causation and correlation. Correlation indicates a relationship between two variables, but it does not imply that one causes the other to happen.
In our baseball example, even though there is a positive correlation between runs scored and attendance, it doesn't prove that scoring more runs causes more fans to attend. Several other factors might be at play:

• Lurking Variables: Factors such as team popularity, day of the week, or special events may influence both runs and attendance.
• Other Influences: Weather conditions, ticket prices, or promotional giveaways could also affect attendance.

Therefore, to confidently say that one variable causes another requires more in-depth analysis beyond simple correlation, often with controlled experiments or longitudinal studies.