Given the following pairs of data: $(1,1)$, $(-1,5)$, and $(10,?)$, which of the following values could represent the correlation coefficient between the $x$ and $y$ variables?

• A. $-1$
• B. $1$
• C. $0.8$
• D. $0$

Answer: (C)

Verified step by step guidance

1. Step 1: Understand that the correlation coefficient measures the strength and direction of a linear relationship between two variables, ranging from -1 to 1.
2. Step 2: Identify the given data points: (1, 1), (-1, 5), and (10, y), where y is unknown. The goal is to find a value of y that could produce a correlation coefficient close to the given options.
3. Step 3: Recall the formula for the Pearson correlation coefficient \(r\): \[r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}\] where \(\bar{x}\) and \(\bar{y}\) are the means of the x and y values respectively.
4. Step 4: Calculate the means \(\bar{x}\) and \(\bar{y}\) using the three x-values (1, -1, 10) and the three y-values (1, 5, y). Express \(\bar{y}\) in terms of y.
5. Step 5: Substitute the values into the correlation formula and solve for y such that the correlation coefficient \(r\) is approximately 0.8, verifying that this value is possible given the data.