The computer-science aptitude score, x, and the achievement score, y (measured by a comprehensive final), were measured for 10 students in a beginning computer-science course. The results were as follows. (Give your answers correct to two decimal places.)
x 7 10 19 9 20 22 6 13 18 15
y 29 27 28 33 24 15 25 23 21 18
(a) Find the equation of the line of best fit.
yhat = + x
(b) Find se2.
To find the equation of the line of best fit (a), we need to compute the values for the regression coefficients (slope and intercept). The equation for the line of best fit is given by:
yhat = b0 + b1 * x
where yhat represents the predicted value of y, b0 represents the y-intercept, and b1 represents the regression coefficient.
To find the values for b0 and b1, we can use linear regression. The formula for b1 (the slope) is given by:
b1 = (n * Σ(xy) - ΣxΣy) / (n * Σ(x^2) - (Σx)^2)
where n is the number of data points, Σxy is the sum of the product of x and y, Σx is the sum of x values, Σy is the sum of y values, and Σ(x^2) is the sum of the squares of x values.
The formula for b0 (the y-intercept) is given by:
b0 = (Σy - b1 * Σx) / n
Using the given values, we can calculate the required sums as follows:
Σx = 7 + 10 + 19 + 9 + 20 + 22 + 6 + 13 + 18 + 15 = 139
Σy = 29 + 27 + 28 + 33 + 24 + 15 + 25 + 23 + 21 + 18 = 243
Σ(x^2) = 7^2 + 10^2 + 19^2 + 9^2 + 20^2 + 22^2 + 6^2 + 13^2 + 18^2 + 15^2 = 2120
Σ(xy) = (7 * 29) + (10 * 27) + (19 * 28) + (9 * 33) + (20 * 24) + (22 * 15) + (6 * 25) + (13 * 23) + (18 * 21) + (15 * 18) = 3546
Now, we can substitute these values into the formulas to calculate b1 and b0:
b1 = (10 * 3546 - 139 * 243) / (10 * 2120 - (139)^2)
b0 = (243 - b1 * 139) / 10
Simplifying these expressions will give us the values for b1 and b0, which we can substitute into the equation yhat = b0 + b1 * x to get the equation of the line of best fit.
For part (b), we need to find se2, which represents the mean squared error (MSE). The formula for MSE is given by:
se2 = Σ(y - yhat)^2 / (n - 2)
To calculate se2, we need to calculate yhat for each x value, subtract yhat from y, square the difference, and sum up these squared differences. Finally, divide by (n - 2) to calculate MSE.
Now let's calculate all these values step by step.