Given the following data for variables $x$ and $y$: $x$: $2$, $4$, $6$, $8$; $y$: $3$, $6$, $7$, $10$. What is the value of the Pearson correlation coefficient $r$ between $x$ and $y$?

• A. $0.00$
• B. $-0.98$
• C. $0.50$
• D. $0.98$

Answer: (D)

Verified step by step guidance

1. First, calculate the mean (average) of the x-values and the mean of the y-values. Use the formulas: \(\bar{x} = \frac{\sum x_i}{n}\) and \(\bar{y} = \frac{\sum y_i}{n}\), where \(n\) is the number of data points.
2. Next, compute the deviations of each x and y value from their respective means: \((x_i - \bar{x})\) and \((y_i - \bar{y})\) for each data point.
3. Then, calculate the covariance between x and y using the formula: \(\text{Cov}(x,y) = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{n-1}\).
4. After that, find the standard deviations of x and y separately using: \(s_x = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}\) and \(s_y = \sqrt{\frac{\sum (y_i - \bar{y})^2}{n-1}}\).
5. Finally, calculate the Pearson correlation coefficient \(r\) by dividing the covariance by the product of the standard deviations: \(r = \frac{\text{Cov}(x,y)}{s_x s_y}\). This value will indicate the strength and direction of the linear relationship between x and y.