Question 20

Please round all answers to 2 decimal places.
Given the table below, use your calculator to find the Correlation Coefficient
\begin{tabular}{|l|r|r|r|r|r|r|}
\hline$x$ & 2.5 & 8.7 & 7.7 & 5.6 & 2.9 & 6.4 \\
\hline$y$ & 7.5 & 19.1 & 17 & 12.1 & 8.2 & 16.3 \\
\hline
\end{tabular}

Put your answer here:

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Step 1 ：Given the data pairs (x, y) as follows: (2.5, 7.5), (8.7, 19.1), (7.7, 17), (5.6, 12.1), (2.9, 8.2), (6.4, 16.3).

Step 2 ：Calculate the sum of x, which is \(2.5 + 8.7 + 7.7 + 5.6 + 2.9 + 6.4 = 33.8\).

Step 3 ：Calculate the sum of y, which is \(7.5 + 19.1 + 17 + 12.1 + 8.2 + 16.3 = 80.2\).

Step 4 ：Calculate the sum of xy, which is \(2.5*7.5 + 8.7*19.1 + 7.7*17 + 5.6*12.1 + 2.9*8.2 + 6.4*16.3 = 511.68\).

Step 5 ：Calculate the sum of \(x^2\), which is \(2.5^2 + 8.7^2 + 7.7^2 + 5.6^2 + 2.9^2 + 6.4^2 = 221.96\).

Step 6 ：Calculate the sum of \(y^2\), which is \(7.5^2 + 19.1^2 + 17^2 + 12.1^2 + 8.2^2 + 16.3^2 = 1189.4\).

Step 7 ：Substitute these values into the correlation coefficient formula: \(r = \frac{n \cdot sum_{xy} - sum_x \cdot sum_y}{\sqrt{(n \cdot sum_{x^2} - sum_x^2) \cdot (n \cdot sum_{y^2} - sum_y^2)}}\), where n is the number of pairs of scores, which is 6 in this case.

Step 8 ：Substitute the values into the formula to get \(r = \frac{6 \cdot 511.68 - 33.8 \cdot 80.2}{\sqrt{(6 \cdot 221.96 - 33.8^2) \cdot (6 \cdot 1189.4 - 80.2^2)}}\).

Step 9 ：Solve the equation to get the correlation coefficient r, which is approximately 0.98.

Step 10 ：Final Answer: The correlation coefficient is \(\boxed{0.98}\).