In a linear regression model y = a + bx, the coefficient b represents the change in y for every unit increase in x. Here, b = 7.25. If the advertising budget (x) increases by $30,000, the predicted change in sales (y) is 7.25 multiplied by 30,000, which equals $217,500.

The slope coefficient is -0.34. If x decreases by 10 units, the change in y is calculated as -0.34 * (-10) = +3.4. Thus, a 10% decrease in the return rate results in a 3.4% increase in the immigration rate, confirming the inverse relationship defined by the negative slope.

Regression analysis models the relationship between variables regardless of their units of measurement. The dependent variable can represent any quantity, such as currency, time, or distance, and does not need to share the same units as the independent variable. The regression coefficients will simply adjust to account for the scale differences.

The provided answer suggests the sign changes, but in linear regression, changing units of measurement (scaling) typically only affects the magnitude of the slope coefficient, not its sign. The sign represents the direction of the relationship, which remains constant regardless of the unit scale. This answer appears factually incorrect based on standard statistical theory.

The regression method of estimation in survey sampling, which uses auxiliary variables to improve precision, is often associated with the work of G.S. Watson. While many statisticians contributed to regression theory, Watson is specifically recognized for his developments in the context of survey estimation techniques.

The two regression coefficients, bxy and byx, must always have the same sign. This is because both coefficients are related to the correlation coefficient r by the formula r = ±√(bxy * byx). Since the square root of a product must be real, the product bxy * byx must be positive, which only occurs if both coefficients are positive or both are negative.

When the regression lines of y on x and x on y coincide, it implies a perfect linear relationship. However, the provided answer key suggests 'Zero'. This is factually contradictory, as coincidence usually implies r = ±1. We retain the provided answer key while noting the potential error in the source material.

The coefficient of determination r^2 represents the proportion of variance explained by the model. If r^2 = 1, the model explains 100% of the variance, meaning the residuals are zero, which implies the Sum of Squared Errors (SSE) must be zero.

The coefficient of determination R-squared is r squared = 0.85 squared = 0.7225. The explained sum of squares (SSR) is R-squared multiplied by SST, which is 0.7225 * 133452 = 96419.07. The residual sum of squares (SSE) is calculated as SST - SSR, resulting in 133452 - 96419.07 = 37032.93. The provided answer B appears to be mathematically inconsistent with standard regression formulas.

In statistical notation, lowercase letters like 'b' are commonly used to represent sample statistics, while uppercase Greek or Latin letters like 'B' (or beta) are used to represent the corresponding population parameters. Thus, 'b' denotes the sample regression coefficient, and 'B' denotes the population regression coefficient.