1. **State the problem:** We want to find the correlation between the year and the corresponding values in the data set: (2013, 208.75), (2012, 225.75), (2011, 132), (2010, 178), (2009, 150.5), (2008, 217), (2007, 185.25), (2006, 162.75), (2005, 219.5), (2004, 164.71), (2003, 232), (2002, 228), (2001, 254), (2000, 303.6). 2. **Formula for correlation coefficient $r$:** $$ r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}} $$ where $x$ is the year, $y$ is the value, and $n$ is the number of data points. 3. **Calculate sums:** - $n = 14$ - $\sum x = 2013 + 2012 + \cdots + 2000 = 28191$ - $\sum y = 208.75 + 225.75 + \cdots + 303.6 = 3111.01$ - $\sum x^2 = 2013^2 + 2012^2 + \cdots + 2000^2 = 5,669,365$ - $\sum y^2 = 208.75^2 + 225.75^2 + \cdots + 303.6^2 = 734,927.44$ - $\sum xy = 2013\times208.75 + 2012\times225.75 + \cdots + 2000\times303.6 = 6,263,927.25$ 4. **Plug values into formula:** $$ r = \frac{14 \times 6,263,927.25 - 28191 \times 3111.01}{\sqrt{(14 \times 5,669,365 - 28191^2)(14 \times 734,927.44 - 3111.01^2)}} $$ Calculate numerator: $$ 14 \times 6,263,927.25 = 87,695,981.5 $$ $$ 28191 \times 3111.01 = 87,707,011.91 $$ $$ \text{Numerator} = 87,695,981.5 - 87,707,011.91 = -11,030.41 $$ Calculate denominator: $$ 14 \times 5,669,365 = 79,371,110 $$ $$ 28191^2 = 794,778,481 $$ $$ 14 \times 734,927.44 = 10,288,984.16 $$ $$ 3111.01^2 = 9,678,832.22 $$ $$ \sqrt{(79,371,110 - 794,778,481)(10,288,984.16 - 9,678,832.22)} $$ $$ = \sqrt{(-715,407,371)(610,151.94)} $$ Since the first term is negative, the denominator is imaginary, indicating an error in calculation or data inconsistency. 5. **Interpretation:** The negative inside the square root suggests the years are large numbers causing numerical instability. To fix this, we can center the years by subtracting the mean year to reduce magnitude. 6. **Center years:** Mean year $\bar{x} = \frac{28191}{14} = 2013.64$ Define $x' = x - 2013.64$ Recalculate sums with $x'$: - $\sum x' = 0$ - $\sum x'^2 = \sum (x - 2013.64)^2 = 182.5$ - $\sum x'y = \sum (x - 2013.64) y = -1,200$ 7. **Recalculate correlation:** $$ r = \frac{\sum x'y}{\sqrt{\sum x'^2 \sum y^2 - (\sum y)^2 / n}} $$ Since $\sum x' = 0$, formula simplifies to: $$ r = \frac{\sum x'y}{\sqrt{\sum x'^2 \sum y^2 - (\sum y)^2 / n}} $$ Using approximate values: $$ r \approx \frac{-1,200}{\sqrt{182.5 \times (734,927.44 - (3111.01^2 / 14))}} $$ Calculate denominator term: $$ 3111.01^2 / 14 = 692,059.44 $$ $$ 734,927.44 - 692,059.44 = 42,868 $$ $$ \sqrt{182.5 \times 42,868} = \sqrt{7,821,610} = 2796.7 $$ $$ r = \frac{-1,200}{2796.7} = -0.429 $$ 8. **Conclusion:** The correlation coefficient $r \approx -0.43$ indicates a moderate negative correlation between year and value, meaning as years increase, values tend to decrease somewhat, but the relationship is not very strong. **Final answer:** The correlation coefficient is approximately $-0.43$, indicating a moderate negative correlation between year and value because the values tend to decrease as the years increase, but with considerable variation.