In regression analysis, the residual term represents the difference between the observed value and the predicted value. It is frequently referred to as the error term, which accounts for unobserved factors, or the disturbance term, which reflects random fluctuations in the data. Both terms are used interchangeably in econometrics and statistics to describe the unexplained portion of the dependent variable.

The relationship between an observed value, the predicted value, and the error term is defined as Observed Value = Predicted Value + Error. Rearranging this formula to solve for the predicted value gives: Predicted Value = Observed Value - Error. Substituting the given figures, 85 - 25 equals 60. This calculation helps in evaluating the accuracy of the regression model.

The probable error (P.E.) of the correlation coefficient is used to determine the reliability of the sample correlation coefficient. The range defined by r ± P.E. provides the limits within which the population correlation coefficient is expected to lie with a certain level of confidence, helping researchers assess the significance of the observed correlation.

Using the formula R = 1 - (6 * Σd²) / (n * (n² - 1)), we substitute the values: R = 1 - (6 * 50) / (10 * (100 - 1)) = 1 - 300 / 990 = 1 - 0.303 = 0.6969. Rounding to two decimal places, we get 0.70.

Correlation does not imply causation. A strong linear relationship between two variables indicates that they move together, but it does not prove that one variable directly causes changes in the other. There could be a third variable influencing both, or the relationship could be purely coincidental.

The Pearson correlation coefficient, denoted as 'r', measures the strength and direction of a linear relationship between two variables. Its value always falls within the range of -1.0 to +1.0, where -1.0 indicates a perfect negative linear correlation, +1.0 indicates a perfect positive linear correlation, and 0 indicates no linear correlation.

A downward-sloping straight line in a scatter diagram indicates that as one variable increases, the other decreases at a constant rate, which represents a perfect negative correlation (r = -1).

The correlation coefficient 'r' ranges from -1 to +1. A perfect positive correlation is represented by +1 (a-2). A perfect negative correlation is represented by -1 (b-3). No correlation between variables is indicated by a coefficient of 0 (c-1). Therefore, the correct mapping is a-2, b-3, and c-1.

The correlation coefficient is independent of the change of origin and scale. Multiplying by a positive constant does not change the sign, but multiplying by a negative constant flips it. Here, -x (negative) and 2y (positive) results in a net negative sign change. However, the provided answer is C. This may be due to a specific interpretation of the variables.

Both statements are mathematically correct. The product of the two regression coefficients (bxy * byx) equals the square of the correlation coefficient (r^2). Since r^2 must be between 0 and 1, if one coefficient is greater than 1, the other must be less than 1 to keep the product within the valid range. Furthermore, the geometric mean of these two coefficients is indeed equal to the correlation coefficient, confirming the validity of the reason provided.