Question

Does a team's batting average depend in any way on the number of home runs hit by the team? The data in the table show the number of team home runs and the overall team batting average for eight selected major league teams for the 2006 season. \(^{14}\) $$ \begin{array}{lcc} \text { Team } & \text { Total Home Runs } & \text { Team Batting Average } \\\ \hline \text { Atlanta Braves } & 222 & .270 \\ \text { Baltimore Orioles } & 164 & .227 \\ \text { Boston Red Sox } & 192 & .269 \\ \text { Chicago White Sox } & 236 & .280 \\ \text { Houston Astros } & 174 & .255 \\ \text { Philadelphia Phillies } & 216 & .267 \\ \text { New York Giants } & 163 & .259 \\ \text { Seattle Mariners } & 172 & .272 \end{array} $$ a. Plot the points using a scatterplot. Does it appear that there is any relationship between total home runs and team batting average? b. Is there a significant positive correlation between total home runs and team batting average? Test at the \(5 \%\) level of significance. c. Do you think that the relationship between these two variables would be different if we had looked at the entire set of major league franchises?

Short answer

If we had data for all major league franchises, do you think the relationship between these two variables might be different?

Step-by-step solution

Plotting the Scatterplot

To visualize the relationship between total home runs and team batting average, we should create a scatterplot with the 'Total Home Runs' on the x-axis and 'Team Batting Average' on the y-axis. Plot the given data points for each team, and observe if there is any pattern or relationship between the two variables.

Describing the Scatterplot

If the points in the scatterplot seem to show a pattern where higher 'Total Home Runs' correspond to higher 'Team Batting Average', it indicates a positive relationship between the two variables. If there is no clear pattern, it might suggest that there is no significant relationship between the two variables.

Calculating the Correlation Coefficient

To test the strength of the relationship between the number of total home runs and team batting average, we need to calculate the correlation coefficient, denoted as 'r'. To do this, use the formula: $$ r = \frac{\Sigma{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\Sigma{(x_i - \bar{x})^2}\Sigma{(y_i - \bar{y})^2}}} $$ where \(x_i\) and \(y_i\) are the individual data points for total home runs and team batting average, respectively, and \(\bar{x}\) and \(\bar{y}\) are their respective means. Calculate the correlation coefficient for the given data.

Checking Significance of Correlation

Once we have the correlation coefficient, we can test for the significance of the correlation at the 5% significance level. To do this, first calculate the test statistic: $$ t = r\sqrt{\frac{n-2}{1-r^2}} $$ where 'n' is the number of data points (teams) and 'r' is the correlation coefficient. Compare the calculated test statistic 't' to the critical value from the t-distribution with 'n-2' degrees of freedom and a 5% significance level. If the calculated test statistic is greater than the critical value, we reject the null hypothesis and conclude that there is a significant positive correlation between total home runs and team batting average.

Discussing Effect of Complete Dataset

Finally, speculate on whether the relationship between total home runs and team batting average might be different if we had data for all major league franchises. It might be possible that the relationship would become stronger, weaker, or stay the same. Discuss factors that could influence the relationship between these two variables.

Key concepts

Scatterplot

In the vast field of statistics, visual representations are essential for identifying potential relationships between variables. The scatterplot is a prime tool for this purpose, often used as a preliminary step in data analysis. To create a scatterplot, you plot pairs of numerical data with one variable on each axis, allowing you to observe potential correlations visually.

For example, if we plot the 'Total Home Runs' on the x-axis and the 'Team Batting Average' on the y-axis based on the exercise data, each major league team represents a point where these two measures intersect. Observing the pattern formed by these points provides an immediate impression of whether a relationship might exist. A upward trend as you move from left to right would suggest a positive correlation—that teams with more home runs tend to have higher batting averages. Conversely, a downward trend would suggest a negative correlation, and a lack of any discernible pattern suggests no correlation at all.

The scatterplot serves as a springboard for further statistical analysis, paving the way for calculating precise measures of correlation and testing their significance.

Correlation Coefficient

Moving beyond visual conjecture from scatterplots, the correlation coefficient, denoted as 'r', provides a quantitative measure of the strength and direction of the linear relationship between two variables. It is a value between -1 and 1 where 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation.

The formula for the correlation coefficient is: \[ r = \frac{\Sigma{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\Sigma{(x_i - \bar{x})^2}\Sigma{(y_i - \bar{y})^2}}} \] Here, \( x_i \) and \( y_i \) are individual data points for total home runs and team batting average, respectively, while \( \bar{x} \) and \( \bar{y} \) represent their means. Before reaching for the calculator, it's important to note that calculating 'r' offers more than just a number—it provides a foundation for making predictions, controlling for variables, and even formulating strategies based on identified trends.

Significance Testing

Establishing a correlation is just the first step; the next critical move is to determine whether it's a result of an actual relationship or merely a coincidence. This is where significance testing marches in, a quintessential process in statistics to confirm the reliability of the results. Specifically, when measuring correlation, we perform significance testing to rule out the likelihood that the observed relationship happened by chance alone.

The primary approach is to use the calculated correlation coefficient ('r') and apply it to a formula that generates a test statistic, given as: \[ t = r\sqrt{\frac{n-2}{1-r^2}} \] Here, 'n' symbolizes the number of data points. We then compare the calculated 't' with a critical value from a specific t-distribution corresponding to 'n-2' degrees of freedom and a predefined significance level—commonly 5%. If our 't' exceeds the critical value, the correlation is deemed significant, and we conclude that it's unlikely the observed relationship is due to random chance. This pivotal step reassures us that our findings are robust enough to warrant further attention, potentially influencing decisions in fields as diverse as sports management and business analytics.