Suppose you are given the following data for two variables, $X$ and $Y$: $(1,2)$, $(2,4)$, $(3,6)$, $(4,8)$. What is the value of the sample correlation coefficient between $X$ and $Y$? Do not round your intermediate calculations.

• A. $0$
• B. $-1$
• C. $1$
• D. $0.5$

Answer: (C)

Verified step by step guidance

1. Identify the pairs of data points for variables X and Y: (1, 2), (2, 4), (3, 6), (4, 8).
2. Calculate the mean of X and the mean of Y using the formulas: \(\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i\) and \(\bar{Y} = \frac{1}{n} \sum_{i=1}^n Y_i\), where \(n\) is the number of data points.
3. Compute the deviations from the mean for each variable: \((X_i - \bar{X})\) and \((Y_i - \bar{Y})\) for each data point.
4. Calculate the covariance between X and Y using the formula: \(\text{Cov}(X,Y) = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})(Y_i - \bar{Y})\).
5. Calculate the sample standard deviations of X and Y: \(s_X = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2}\) and \(s_Y = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (Y_i - \bar{Y})^2}\). Then, find the sample correlation coefficient \(r\) using the formula: \(r = \frac{\text{Cov}(X,Y)}{s_X s_Y}\).