Questions: Question 10 (1 point) 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, 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. No, you should examine the situation to identify lurking variables that may be influencing both variables.

Solution Steps

The covariance between the variables \( X \) and \( Y \) is calculated as: \[ \text{Cov}(X,Y) = 5.0 \] The standard deviations 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 linear regression results are: \[ \text{Linear Regression Result: } \{ \text{correlation_coefficient}: 1.0, \alpha: 0.0, \beta: 2.0 \} \]

Despite the strong linear relationship indicated by the correlation coefficient \( r = 1.0 \), we cannot claim causation. It is essential to consider potential lurking variables that may influence both \( X \) and \( Y \).

Final Answer