The coefficient of determination, denoted as R-squared, measures the proportion of the variance in the dependent variable that is predictable from the independent variable. A higher R-squared value indicates that the data points are clustered more tightly around the regression line, which is constructed using the method of least squares to minimize the sum of squared residuals.

Correlation can arise from a direct causal link, a mutual reaction between variables, or the influence of a third common variable (spurious correlation). Since all listed options are recognized potential causes of observed correlation, 'None of these' is the correct choice as none of the listed items are 'not' a cause.

The correlation coefficient between two variables X and Y is symmetric. This means that the correlation of X with Y is identical to the correlation of Y with X. Therefore, γxy is equal to γyx. This property holds because the formula for correlation involves the product of deviations from the mean, which is commutative.

The multiple correlation coefficient ranges from 0 to 1. A value of 1 indicates that the dependent variable can be perfectly predicted by the linear combination of the independent variables, signifying a perfect linear relationship between the observed and predicted values.

The coefficient of determination, denoted as R-squared, represents the proportion of the variance in the dependent variable that is predictable from the independent variable. The correlation coefficient (r) is the square root of this value, though it retains the sign of the relationship, unlike R-squared which is always non-negative.

The correlation coefficient is a dimensionless measure of linear association. It is invariant to linear transformations of the variables, such as changes in units of measurement (scaling). Since converting feet to inches and pounds to ounces involves only multiplication by positive constants, the correlation coefficient remains unchanged at 0.40.

The coefficient of determination (R-squared) is the square of the correlation coefficient (r). Here, r = 0.8, so R-squared = 0.64, which represents 64% of the variance. The provided answer key selects 80%, which is incorrect as it represents the correlation coefficient itself rather than the explained variation.

The Pearson correlation coefficient, denoted by r, measures the strength and direction of a linear relationship between two variables. Its mathematical range is bounded between -1 and +1, inclusive, where -1 indicates a perfect negative linear relationship and +1 indicates a perfect positive linear relationship.

The 95% confidence interval for a proportion is calculated using the formula P ± Z * sqrt(pq/n), where Z is the critical value for a 95% confidence level (approximately 1.96). The provided option D represents the standard form of this calculation, where p is the sample proportion, q is 1-p, and n is the sample size. Note that the notation in the options is slightly simplified.

To calculate the 90% confidence interval for a proportion, use the formula p ± z*(sqrt(p(1-p)/n)). With p=0.69, n=400, and z=1.645, the margin of error is approximately 0.038. This results in an interval of [0.652, 0.728]. The provided answer matches this calculation, confirming the interval boundaries.