Questions: Question 11 (2 points) Suppose you are interested in the correlation between two variables. The correlation between those two variables will be the further a set of data points fall from the regression line. stronger less reliable weaker more reliable

Solution Steps

The means of the variables \( X \) and \( Y \) are calculated as follows:

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 3.0 \]

\[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 4.0 \]

The correlation coefficient \( r \) is computed using the formula:

\[ r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \]

Where:

• Covariance \( \text{Cov}(X,Y) = 1.5 \)
• Standard deviation of \( X \), \( \sigma_X = 1.5811 \)
• Standard deviation of \( Y \), \( \sigma_Y = 1.2247 \)

Substituting the values:

\[ r = \frac{1.5}{1.5811 \times 1.2247} = 0.7746 \]

The slope \( \beta \) and intercept \( \alpha \) of the regression line are calculated as follows:

1. Numerator for \( \beta \):

\[ \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 66 - 5 \times 3.0 \times 4.0 = 6.0 \]

2. Denominator for \( \beta \):

\[ \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 55 - 5 \times 3.0^2 = 10.0 \]

3. Slope \( \beta \):

\[ \beta = \frac{6.0}{10.0} = 0.6 \]

4. Intercept \( \alpha \):

\[ \alpha = \bar{y} - \beta \bar{x} = 4.0 - 0.6 \times 3.0 = 2.2 \]

The equation of the line of best fit is given by:

\[ y = 2.2 + 0.6x \]

The correlation coefficient \( r = 0.7746 \) indicates a strong positive correlation. Therefore, the correlation between the two variables will be weaker the further a set of data points fall from the regression line.

Final Answer