Questions: Compute the linear correlation coefficient for the data below, rounding to three decimal places.

Solution Steps

To compute the linear correlation coefficient (also known as Pearson's correlation coefficient) for a given set of data, follow these steps:

1. Calculate the mean of the x-values and the mean of the y-values.
2. Compute the covariance of the x and y values.
3. Calculate the standard deviations of the x-values and y-values.
4. Use the formula for the Pearson correlation coefficient: \( r = \frac{cov(X, Y)}{\sigma_X \sigma_Y} \).

The mean of the \( x \)-values is calculated as: \[ \text{mean}_x = \frac{10 + 20 + 30 + 40 + 50}{5} = 30.0 \] The mean of the \( y \)-values is calculated as: \[ \text{mean}_y = \frac{15 + 25 + 35 + 45 + 55}{5} = 35.0 \]

The covariance between \( x \) and \( y \) is calculated as: \[ \text{cov}(X, Y) = \frac{1}{n} \sum_{i=1}^{n} (x_i - \text{mean}_x)(y_i - \text{mean}_y) \] Given the values, the covariance is: \[ \text{cov}(X, Y) = 200.0 \]

The standard deviation of the \( x \)-values is: \[ \sigma_X = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \text{mean}_x)^2} = 14.1421 \] The standard deviation of the \( y \)-values is: \[ \sigma_Y = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (y_i - \text{mean}_y)^2} = 14.1421 \]

The Pearson correlation coefficient \( r \) is calculated as: \[ r = \frac{\text{cov}(X, Y)}{\sigma_X \sigma_Y} = \frac{200.0}{14.1421 \times 14.1421} = 1.0 \]

Final Answer