The coefficient of determination R-squared is defined as the ratio of the Sum of Squares Regression (SSR) to the Total Sum of Squares (SST). If R-squared equals 1, it implies that the regression model explains all the variability in the dependent variable. Therefore, the explained sum of squares (SSR) must be equal to the total sum of squares (SST).

The regression coefficients are defined as bxy = r * (σx/σy) and byx = r * (σy/σx). If bxy = byx = r, then the ratios of the standard deviations must be equal to 1. This implies that σx/σy = 1 and σy/σx = 1, which leads to the conclusion that the standard deviation of X must be equal to the standard deviation of Y.

Sir Francis Galton, a pioneering statistician and polymath, first coined the term 'regression' in the late 19th century. He observed that the heights of children born to exceptionally tall or short parents tended to 'regress' or move back toward the population average, a phenomenon he termed 'regression toward mediocrity'.

Variables that influence the outcome are known by several names depending on the context. 'Independent variable' is common in general research, 'predictor variable' is standard in regression analysis, and 'controlled variable' is used when the researcher actively manipulates the variable to observe its effect on the response.

A perfect estimator implies that the predicted values exactly match the observed values. This occurs when all data points lie directly on the regression line, resulting in a residual sum of squares of zero. Consequently, the standard error of the estimate, which measures the average deviation from the line, must also be zero.

The provided answer suggests that a regression line is evidence of a cause-effect relationship. Statistically, correlation or regression does not imply causation; it only shows association. While the answer key claims B, this is a common misconception in statistics. We preserve the key as requested.

Plugging x = 5 into the equation: y = -0.0127 + 0.0180(5) = -0.0127 + 0.0900 = 0.0773. Comparing this to the legal limit of 0.08, the difference is 0.08 - 0.0773 = 0.0027. Thus, the predicted value is 0.0027 below the legal limit.

In regression analysis, the variable used to predict or explain the variation in the dependent variable is known by several interchangeable terms: the independent variable, the predictor variable, or simply the x-variable in the context of the regression equation.

The relationship between regression coefficients is defined by b_xy * b_yx = r^2. Since the square of the correlation coefficient r^2 is always between 0 and 1, if b_xy > 1, then b_yx must be less than 1 to ensure their product remains within the range [0, 1].

Regression analysis between two variables X and Y allows for two distinct models: regressing Y on X (predicting Y based on X) and regressing X on Y (predicting X based on Y). Thus, estimation is considered two-way.