Based on the available information, the correct option is (d) y = -2x.
T determine the correct regression equation based on the given information, consider the relationship between the variables x and y, as well as the variance of x.
In a linear regression equation of the form y = mx + b, the slope (m) represents the relationship between the variables. The variance of x is related to the slope as follows:
variance of x = (1 / n) * Σ((x - X)²)
where n is the number of data points, x is the individual data point, and X is the mean of x.
From the given information, the variance of x is 9. This means:
9 = (1 / n) * Σ((x - X)²)
Since the variance is positive, the sum of squared differences must also be positive. This implies that the values of x are not all the same.
Now, analyze the options:
(a) y = 3x: This option does not provide any information about the variance of x.
(b) y = 2x: This option also does not provide any information about the variance of x.
(c) y = -3x: This option does not match the given information.
(d) y = -2x: This option matches the given information. The slope is -2, which means that for every unit increase in x, y decreases by 2. This equation is consistent with the variance of x being 9.
Therefore, the correct option is (d) y = -2x.