QUESTION 9

Provide an appropriate response.
In an area of the Great Plains, records were kept on the relationship between the rainfall (in inches) and the yield of wheat (bushels per acre). Calculate the linear correlation coefficient.
\begin{tabular}{l|c|c|c|c|c|c|c|c|c|}
Rainfall (in inches), $x$ & 13.4 & 11.7 & 16.3 & 15.4 & 21.7 & 13.2 & 9.9 & 18.5 & 18.9 \\
\hline Yield (bushels per acre), $y$ & 51.5 & 47.2 & 59.8 & 60 & 83.4 & 50.2 & 32.9 & 77 & 79.8
\end{tabular}
0.981
0.900
0.998
0.899
10 points
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Step 1 ：Calculate the sum of \( x \) values: \( \sum x = 139.0 \)

Step 2 ：Calculate the sum of \( y \) values: \( \sum y = 541.8 \)

Step 3 ：Calculate the sum of the product of corresponding \( x \) and \( y \) values: \( \sum xy = 8871.93 \)

Step 4 ：Calculate the sum of the squares of \( x \) values: \( \sum x^2 = 2261.9 \)

Step 5 ：Calculate the sum of the squares of \( y \) values: \( \sum y^2 = 34911.18 \)

Step 6 ：Use the formula for Pearson's correlation coefficient: \( r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \)

Step 7 ：Substitute the values into the formula: \( r = \frac{9(8871.93) - (139.0)(541.8)}{\sqrt{[9(2261.9) - (139.0)^2][9(34911.18) - (541.8)^2]}} \)

Step 8 ：Calculate the linear correlation coefficient: \( r = 0.9808184962339461 \)

Step 9 ：Round the result to three decimal places: \( r = 0.981 \)

Step 10 ：Final Answer: \(\boxed{0.981}\)