Question

The basketball coach at a small local college believes that his team plays better and scores more points in front of larger crowds. The number of points scored and attendance for all home games last season are reported here. Do these data support the coach's argument? $$ \begin{array}{ccc} \hline \text { Game } & \text { Points Scored } & \text { Attendance } \\ \hline 1 & 54 & 378 \\ 2 & 57 & 350 \\ 3 & 59 & 320 \\ 4 & 80 & 478 \\ 5 & 82 & 451 \\ 6 & 75 & 250 \\ 7 & 73 & 489 \\ 8 & 53 & 451 \\ 9 & 67 & 410 \\ 10 & 78 & 215 \\ 11 & 67 & 113 \\ 12 & 56 & 250 \\ 13 & 85 & 450 \\ 14 & 101 & 489 \\ 15 & 99 & 472 \end{array} $$

Short answer

Calculate the correlation coefficient to see if it's significantly positive.

Step-by-step solution

Organize the Data

List the points scored and attendance data to see the relationship between the number of points and the crowd size.

Calculate the Correlation Coefficient

Use the Pearson correlation coefficient formula: \[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n(\sum x^2) - (\sum x)^2][n(\sum y^2) - (\sum y)^2]}} \] Where \(n\) is the number of games, \(x\) is attendance, and \(y\) is points scored.

Perform the Calculations

1. Find \(\sum x\), \(\sum y\), \(\sum xy\), \(\sum x^2\), and \(\sum y^2\).2. Plug the values into the formula to find the correlation coefficient \(r\).

Interpret the Correlation Coefficient

A positive \(r\) value indicates a positive correlation between attendance and points scored. A value near 1 suggests a strong positive correlation.

Compare Correlation Coefficient to Threshold

Determine if the correlation is statistically significant by comparing it to a critical value for the given sample size. Typically, a correlation is considered significant if \(|r| > 0.5\) for 15 data points.

Conclusion

Based on the correlation coefficient and its significance, decide whether the data supports the coach's argument.

Key concepts

Correlation Coefficient Calculation

In this exercise, we use the Pearson correlation coefficient to analyze if there's a relationship between the points scored and game attendance for a basketball team. The formula is:
\[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n(\sum x^2) - (\sum x)^2][n(\sum y^2) - (\sum y)^2]}} \]
Here,

• \(n\) is the number of games
• \(x\) represents attendance
• \(y\) represents points scored

To start, you need to calculate the following:

• \(\sum x\)
• \(\sum y\)
• \(\sum xy\)
• \(\sum x^2\)
• \(\sum y^2\)

After these sums are determined, you substitute them into the formula to find \(r\). If \(r\) is positive, it indicates a positive correlation, meaning higher attendance might be associated with more points scored. If \(r\) is closer to 1, it reflects a stronger positive correlation.

Data Interpretation in Sports

Interpreting statistical data is crucial in sports analytics. Once we calculate the correlation coefficient in our example, understanding what it means is the next step.
A positive correlation coefficient suggests that as one variable increases, so does the other. In our context, a higher attendance might lead to more points scored.

• Positive Correlation: When fans show up in larger numbers, the team scores more points.
• Negative Correlation: If larger crowds were linked to fewer points scored, we would have a negative correlation.
• No Correlation: A very low or zero correlation coefficient would suggest no relationship between attendance and points scored.

Interpreting this data can assist coaches in making decisions. For example, if a positive correlation is found, the coach might encourage marketing efforts to boost game attendance. Data interpretation hence transforms raw data into actionable insights.

Statistical Significance in Social Research

Statistical significance helps us understand if our findings are likely due to a true effect or just by random chance. In our basketball example, we need to determine if the correlation coefficient \(r\) is statistically significant.
Generally, in social research, a correlation is considered significant if \(|r| > 0.5\) for a decent sample size like 15.
A significant \(r\) indicates that the observed relationship is strong enough not to be due to random variations.

• Threshold Value: Compare \(r\) to a critical value, typically found in statistical tables.
• P-Value: Often used to determine statistical significance, with a value less than 0.05 indicating significance.

If the correlation is statistically significant, it strengthens the coach's argument that larger crowds boost team performance. Statistical significance thus validates our analysis and ensures the findings are reliable and meaningful in a broader context.