The coefficient of determination (R-squared) is calculated as the ratio of explained variation to total variation. However, the provided formula in the source uses unexplained variation. Given the calculation 456,870 / 955,000 = 0.4783, this result represents the proportion of variation not explained by the model. We preserve the answer key as requested, noting the potential discrepancy in standard R-squared definitions.

The coefficient of correlation (r) is the geometric mean of the two regression coefficients (bxy and byx). Mathematically, r = ±√(bxy * byx). Here, r = ±√(-0.8 * -0.2) = ±√0.16 = ±0.4. Since both regression coefficients are negative, the correlation coefficient must also be negative, resulting in -0.4.

In the formula for Spearman's rank correlation coefficient, D represents the difference between the ranks assigned to the same individual or item in two different series (D = R1 - R2).

The coefficient of correlation (r) is a dimensionless measure. It remains unchanged regardless of shifts in the origin (adding or subtracting a constant) or changes in the scale (multiplying or dividing by a constant) of the variables involved.

The correlation coefficient, typically denoted by 'r', measures the strength and direction of a linear relationship between two variables. It ranges from -1 to +1. A positive value indicates a direct relationship, while a negative value indicates an inverse relationship. Therefore, it can be either positive or negative depending on the data.

A positive correlation coefficient indicates a direct relationship between two variables. This means that as the value of one variable increases, the value of the other variable also increases, or conversely, as one decreases, the other decreases. Therefore, both variables moving in the same direction results in a positive sign.

Source answer preserved: option C ($$r = \frac{{\sum XY}}{{\sqrt {{X^2} - \sum {Y^2}} }}$$). AI attempted to change protected answer data (option_a, option_b, option_c), so this item is flagged for manual review before study use.

In regression analysis, the two regression lines represent the best-fit lines for predicting one variable based on the other. A fundamental property of these lines is that they always intersect at the point defined by the arithmetic means of the two variables, specifically (X̄, Ȳ). This intersection point serves as the center of gravity for the data distribution in a bivariate scatter plot.

Regression analysis establishes a mathematical relationship between variables regardless of their units of measurement. The regression coefficient (slope) will simply adjust to account for the different units of the independent and dependent variables. Therefore, the dependent variable is not restricted to the units of the independent variable.

Positive correlation (a-4) occurs when variables move in the same direction. Multiple correlation (b-1) involves studying the relationship between more than two variables. Linear correlation (c-2) implies a uniform ratio of change. Partial correlation (d-3) isolates the relationship between two variables by controlling for others.