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The Pearson correlation coefficient, often denoted as 'r', measures the strength and direction of the linear relationship between two variables. The value of 'r' ranges from -1 to 1.
- A value of 1 indicates a perfect positive linear relationship.
- A value of -1 indicates a perfect negative linear relationship.
- A value of 0 indicates no linear relationship.
In our exercise, the Pearson correlation coefficient is calculated using the 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 data points, x and y are the respective values of the variables being examined. For our sports reporter scenario:
- Sum of x-values (preseason AP ranks) is 161
- Sum of y-values (final AP ranks) is 233
- Sum of x² is 3343
- Sum of y² is 10129
- Sum of xy products is 4973
When we substitute these values into the formula, solving through steps, we get the correlation coefficient of approximately 0.676, indicating a moderate positive relationship between preseason and final AP ranks. This method helps quantify how strongly preseason ranks predict final rankings.

Significance testing helps us determine whether the observed correlation is statistically significant or if it could have occurred by chance. In this exercise, we use a t-test for the correlation coefficient. The formula for the t-test is:
\[ t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}} \] Here, n is the number of data points and r is the Pearson correlation coefficient. Plugging in our values:
\[ t = \frac{0.676 \sqrt{7}}{\sqrt{1-0.676^2}} \ = \frac{0.676 \times 2.646}{\sqrt{1-0.456}} \ = \frac{1.789} {\sqrt{0.544}} \ = \frac{1.789}{0.737} \ \approx 2.429] \] For 9 teams (n=9), the degrees of freedom is 7. Referring to the t-distribution table, the critical value for 7 degrees of freedom at \(\alpha=0.05\) is approximately 2.365.
Since our calculated t-value of 2.429 is greater than 2.365, the correlation is considered significant. This means there is less than a 5% chance that the observed correlation occurred by random chance, confirming a statistically significant positive relationship between preseason and final AP ranks.

A scatter plot is a graphical representation of two variables. It helps us visually inspect the relationship and identify trends or patterns. To create a scatter plot for the given data:
- Plot each team's preseason AP rank on the x-axis.
- Plot each team's final AP rank on the y-axis.
- Draw individual points for each (x, y) pair.
When we plot the points for data pairs like (6, 22), (14, 13), (27, 41), (12, 9), and so forth, we observe an upward trend in the scatter plot.
Even though there are fluctuations, the general trend supports the conclusion that higher preseason ranks tend to correspond with better final ranks. This visual confirmation aligns with our calculated Pearson correlation coefficient and the significance test, together showing a moderate positive relationship.