Questions: You determine there is a strong linear relationship between two variables using a test for linear regression. Can you immediately claim that one variable is causing the second variable to act in a certain way? No, you should examine the situation to identify lurking variables that may be influencing both variables. No, the correlation would need to be a perfect linear relationship to be sure. No, you must first decide if the relationship is positive or negative. Yes, a strong linear relationship implies causation between the two variables.

Solution Steps

The covariance between the variables \( X \) and \( Y \) is calculated as: \[ \text{Cov}(X,Y) = 5.0 \] The standard deviations of \( X \) and \( Y \) are: \[ \sigma_X = 1.5811, \quad \sigma_Y = 3.1623 \] The correlation coefficient \( r \) is given by: \[ r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} = \frac{5.0}{1.5811 \times 3.1623} = 1.0 \] Thus, the results are: \[ \text{Covariance and Correlation Coefficient: } \{ \text{correlation_coefficient_rounded}: 1.0, \text{covariance}: 5.0 \} \]

The means of \( X \) and \( Y \) are calculated as: \[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 3.0, \quad \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 6.0 \] The numerator for the slope \( \beta \) is: \[ \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 110 - 5 \times 3.0 \times 6.0 = 20.0 \] The denominator for the slope \( \beta \) is: \[ \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 55 - 5 \times 3.0^2 = 10.0 \] Thus, the slope \( \beta \) is: \[ \beta = \frac{20.0}{10.0} = 2.0 \] The intercept \( \alpha \) is calculated as: \[ \alpha = \bar{y} - \beta \bar{x} = 6.0 - 2.0 \times 3.0 = 0.0 \] The equation of the line of best fit is: \[ y = 0.0 + 2.0x \] The results of the linear regression are: \[ \text{Linear Regression Result: } \{ \text{correlation_coefficient}: 1.0, \alpha: 0.0, \beta: 2.0 \} \]

A strong linear relationship (high correlation coefficient \( r = 1.0 \)) does not imply causation. Correlation does not imply causation because there could be lurking variables affecting both variables. Therefore, the correct answer to the question is: \[ \text{No, you should examine the situation to identify lurking variables that may be influencing both variables.} \]

Final Answer