3. Two variables, x and y, have a correlation of 0.75. If x has a mean of 25 and a standard deviation of 3, and y has a mean of 12 and a standard deviation of 6, which of the following is the least-squares regression line for the two variables?
(A) y(hat) = -25.5+1.5x
(B) y(hat) = 12+1.5x
(C) y(hat) = 12+0.75x
(D) y(hat) = 16+0.75x
(E) Not enough information
The answer is A...you're welcome :-)
To find the least-squares regression line, we need to use the formula:
y(hat) = b0 + b1x
Where y(hat) is the predicted value of y, b0 is the y-intercept, b1 is the slope, and x is the independent variable.
Given that the correlation between x and y is 0.75, we can use the following formula to find the slope (b1):
b1 = r * (s_y / s_x)
Where r is the correlation coefficient, s_y is the standard deviation of y, and s_x is the standard deviation of x.
Plugging in the given values, we have:
b1 = 0.75 * (6 / 3) = 1.5
Now, to find the y-intercept (b0), we can use the formula:
b0 = y_mean - (b1 * x_mean)
Plugging in the given values, we have:
b0 = 12 - (1.5 * 25) = -25.5
Therefore, the correct least-squares regression line is:
y(hat) = -25.5 + 1.5x
So the correct answer is (A) y(hat) = -25.5+1.5x.
To determine the least-squares regression line for the two variables, we need to find the slope and y-intercept of the line based on the given information.
The slope of the regression line (denoted as "b") can be calculated by dividing the covariance of x and y by the variance of x. The formula for the slope is:
b = Cov(x, y) / Var(x)
The y-intercept of the regression line (denoted as "a") can be calculated by subtracting the product of the slope and the mean of x from the mean of y. The formula for the y-intercept is:
a = Mean(y) - b * Mean(x)
Given that the correlation between x and y is 0.75, we know that the correlation coefficient (denoted as "r") is equal to the ratio of the covariance of x and y to the product of the standard deviations of x and y. The formula for the correlation coefficient is:
r = Cov(x, y) / (SD(x) * SD(y))
We can rearrange this equation to solve for the covariance of x and y:
Cov(x, y) = r * SD(x) * SD(y)
Given the means and standard deviations of x and y, we can now calculate the slope and y-intercept of the least-squares regression line by plugging the values into the formulas.
b = (0.75 * 3 * 6) / (3^2) = 0.5
a = 12 - 0.5 * 25 = -0.5
Finally, we can write the equation of the least-squares regression line:
y(hat) = -0.5 + 0.5x
Comparing this equation to the given options, we can see that the closest match is:
(A) y(hat) = -25.5 + 1.5x
Therefore, the correct answer is (A) y(hat) = -25.5 + 1.5x.