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 6 9 11 11 5 14 8 15 20
y 29 28 32 18 21 24 32 24 24 32
(a) Find the equation of the line of best fit.
y hat = + x
(b) Find se2.
y hat = .255x + 23.695
y bar = 26.4
x bar = 10.6
SE² =
[ Σ(y-yBar)^2 - [Σ(x-xBar)(y-yBar)]²/Σ(x-xBar)² ] /(n-2)
To find the equation of the line of best fit, we can use a method called linear regression. This method finds the equation of the line that best represents the relationship between two variables, in this case, the computer-science aptitude score (x) and the achievement score (y).
(a) The equation of the line of best fit is represented as ŷ = a + bx, where a is the y-intercept and b is the slope of the line.
To find these values, we can follow these steps:
1. Calculate the means (average) of both x and y. Let's denote the mean of x as x̄ and the mean of y as ȳ.
x̄ = (7 + 6 + 9 + 11 + 11 + 5 + 14 + 8 + 15 + 20) / 10 = 10.6
ȳ = (29 + 28 + 32 + 18 + 21 + 24 + 32 + 24 + 24 + 32) / 10 = 25.4
2. Calculate the covariance of x and y. Let's denote the covariance as Cov(x, y).
Cov(x, y) = ( (7 - 10.6) * (29 - 25.4) ) + ( (6 - 10.6) * (28 - 25.4) ) + ... + ( (20 - 10.6) * (32 - 25.4) )
Cov(x, y) = (-3.6 * 3.6) + (-4.6 * 2.6) + ... + (9.4 * 6.6)
Cov(x, y) = -12.96 - 11.96 + ... + 61.64
Cov(x, y) = -99.96
3. Calculate the variance of x. Let's denote the variance as Var(x).
Var(x) = ( (7 - 10.6)^2 ) + ( (6 - 10.6)^2 ) + ... + ( (20 - 10.6)^2 )
Var(x) = ( -3.6^2 ) + ( -4.6^2 ) + ... + ( 9.4^2 )
Var(x) = 74.16 + 21.16 + ... + 88.36
Var(x) = 354.8
4. Calculate the slope (b) of the line.
b = Cov(x, y) / Var(x)
b = -99.96 / 354.8
b ≈ -0.28
5. Calculate the y-intercept (a) of the line.
a = ȳ - (b * x̄)
a = 25.4 - (-0.28 * 10.6)
a ≈ 28.45
Therefore, the equation of the line of best fit is ŷ ≈ 28.45 - 0.28x.
(b) To find se2, which represents the estimated variance of residuals, we can follow these steps:
1. Calculate the sum of squared residuals (SSR), given by:
SSR = ( (y1 - ŷ1)^2 ) + ( (y2 - ŷ2)^2 ) + ... + ( (yn - ŷn)^2 )
2. Calculate the residual standard error squared (se2), given by:
se2 = SSR / (n - 2)
In this case, we have 10 data points, so n = 10.
Substitute the given values into the formulas to calculate se2. Note that ŷi represents the predicted value of y for each corresponding x value.
SSR = ( (29 - (28.45 - 0.28 * 7))^2 ) + ( (28 - (28.45 - 0.28 * 6))^2 ) + ... + ( (32 - (28.45 - 0.28 * 20))^2 )
= (0.45^2) + (-0.13^2) + ... + (-2.55^2)
≈ 21.13
se2 = SSR / (n - 2)
= 21.13 / (10 - 2)
≈ 2.64
Therefore, se2 is approximately 2.64.