Regression lines represent the relationship between variables and must intersect at the point of the means of the variables. Because they share this common intersection point, they cannot be parallel unless the correlation coefficient is zero, in which case they are perpendicular.

The correlation coefficient (r) is defined as the geometric mean of the two regression coefficients, bxy and byx. Mathematically, r = ±√(bxy * byx). Therefore, the square root of the product of these coefficients yields the correlation coefficient.

When the correlation coefficient (r) is zero, the two regression lines (Y on X and X on Y) are perpendicular to each other, meaning they intersect at a 90-degree angle. Since the provided answer key is D, it conflicts with standard statistical theory which dictates they are at 90 degrees. This may be due to a misinterpretation of the options provided.

The method of least squares is a mathematical procedure used to find the best-fitting line through a set of data points. It works by minimizing the sum of the squares of the vertical deviations (residuals) between the observed data points and the fitted regression line.

The disturbance term, or residual, represents the difference between the actual observed value and the value predicted by the regression model. It is calculated as the observed value minus the predicted value. In this case, 62 minus 29 equals 33. This residual captures the portion of the dependent variable that the model fails to explain, reflecting random error or omitted variables in the analysis.

In statistical regression models, the dependent variable is the outcome being predicted. It is commonly referred to as the regressand, the explained variable, or the response variable, while the independent variables are known as regressors or predictors.

Spearman's rank correlation coefficient is calculated based on the differences between ranks. Since the ranks are in perfectly inverse order (1 to 5 vs 5 to 1), the correlation is perfectly negative. Using the formula 1 - (6 * Σd²) / (n(n²-1)), where Σd² = 40, the result is 1 - (240/120) = -1.

The two regression lines, representing the relationship between two variables, always intersect at the point defined by the arithmetic means of the two variables, denoted as (x̄, ȳ). This is a fundamental property of linear regression analysis.

The regression coefficient of x on y, denoted as bxy, represents the slope of the regression line of x on y. It is calculated by multiplying the correlation coefficient (r) by the ratio of the standard deviation of x to the standard deviation of y. This formula quantifies how much x changes for a unit change in y.

Source answer preserved: option B ($$1 - \frac{{6\sum {d^2}}}{{N\left( {{N^2} - 1} \right)}}$$). AI attempted to change protected answer data (option_a, option_b, option_c, option_d), so this item is flagged for manual review before study use.