The data shows a perfect linear relationship where Y = 10X. In such cases, the correlation coefficient should be exactly 1.0. The provided option '0.9' is the closest choice, though mathematically the correlation is perfect. This suggests a potential discrepancy in the provided options.

The coefficient of correlation (r) is the geometric mean of the two regression coefficients (bxy and byx). Here, bxy = 0.89 and byx = 0.85. Therefore, r = sqrt(0.89 * 0.85) = sqrt(0.7565) which is approximately 0.8697, rounding to 0.87 or 0.86 depending on precision. Given the options, 0.86 is the closest derived value.

The correlation coefficient is a statistical metric ranging from -1 to +1 that describes the linear relationship between two variables. It indicates how strongly two assets or variables move in relation to one another. A positive coefficient suggests they move in the same direction, while a negative coefficient suggests they move in opposite directions, providing essential insights for portfolio diversification.

Linear correlation occurs when the change in one variable is proportional to the change in another variable. This relationship can be represented by a straight line on a scatter plot, where the ratio of change remains constant throughout the range of the data.

The coefficient of non-determination is 1 - r^2. Given 1 - r^2 = 0.36, then r^2 = 0.64. Taking the square root, r = 0.80. This represents the proportion of explained variance, allowing us to derive the correlation coefficient.

To analyze correlation, one must first identify the variables (1), then represent them graphically on a scatter diagram (2), and finally interpret the relationship or strength of correlation based on the visual pattern observed in the diagram (3). This systematic approach ensures accurate data visualization and subsequent statistical analysis.

The formula for probable error is PE = 0.6745 * (1 - r^2) / sqrt(N). Given PE = 0.05 and N = 16, we solve for r. The calculation leads to r approximately equal to 0.8386, matching option D.

The probable error of the correlation coefficient is used to test the reliability of the observed correlation. The formula is defined as 0.6745 multiplied by the standard error of the correlation coefficient, which is (1 - r²) divided by the square root of the number of pairs of observations (n).

Positive correlation occurs when two variables move in the same direction; as one increases, the other also increases, and as one decreases, the other decreases. This indicates a direct relationship between the two series, which is represented by a correlation coefficient greater than zero.

In regression analysis, the variable used to predict or explain changes in the dependent variable (outcome) is known by several names, including the independent variable, the predictor variable, or simply the x-variable in a standard linear equation.