Use the following table to answer the questions. (Give your answers correct to two decimal places.)
x 1 1 3 3 5 5 7 7 9 9
y 3 2 6 1 3 3 3 2 5 3
(a) Find the equation of the line of best fit.
y hat = + x
(ii) Graph this equation on a scatter diagram. (Do this on paper. Your instructor may ask you to turn in this work.)
(b) Find the ordinates y hat for the points on the line of best fit whose abscissas are x = 1, 3, 5, 7, and 9.
1 . .
3 . .
5 . .
7 . .
9
(c) Find the value of e for each of the points above (e = y - y hat).
y e y e
3 3. .
2 3
6 2. .
1 5
3 3
(d) Find the variance se2 of those points about the line of best fit.
To find the equation of the line of best fit, we can use the method of least squares. Here are the steps to calculate it:
Step 1: Calculate the mean of x (x̄) and the mean of y (ȳ).
x̄ = (1 + 1 + 3 + 3 + 5 + 5 + 7 + 7 + 9 + 9) / 10 = 5
ȳ = (3 + 2 + 6 + 1 + 3 + 3 + 3 + 2 + 5 + 3) / 10 = 3
Step 2: Calculate the sum of the products of (x - x̄) and (y - ȳ).
Σ[(x - x̄)(y - ȳ)] = (1 - 5)(3 - 3) + (1 - 5)(2 - 3) + (3 - 5)(6 - 3) + (3 - 5)(1 - 3) + (5 - 5)(3 - 3) + (5 - 5)(3 - 3) + (7 - 5)(3 - 3) + (7 - 5)(2 - 3) + (9 - 5)(5 - 3) + (9 - 5)(3 - 3) = -6
Step 3: Calculate the sum of the squares of (x - x̄).
Σ[(x - x̄)²] = (1 - 5)² + (1 - 5)² + (3 - 5)² + (3 - 5)² + (5 - 5)² + (5 - 5)² + (7 - 5)² + (7 - 5)² + (9 - 5)² + (9 - 5)² = 40
Step 4: Calculate the slope of the line (β₁) using the formula:
β₁ = Σ[(x - x̄)(y - ȳ)] / Σ[(x - x̄)²]
β₁ = -6 / 40 = -0.15
Step 5: Calculate the y-intercept of the line (β₀) using the formula:
β₀ = ȳ - β₁ * x̄
β₀ = 3 - (-0.15) * 5 = 3.75
Thus, the equation of the line of best fit is: ŷ = 3.75 - 0.15x
To find the ordinates ŷ for the points on the line of best fit, substitute the given x values into the equation:
For x = 1, ŷ = 3.75 - 0.15 * 1 = 3.60
For x = 3, ŷ = 3.75 - 0.15 * 3 = 3.45
For x = 5, ŷ = 3.75 - 0.15 * 5 = 3.30
For x = 7, ŷ = 3.75 - 0.15 * 7 = 3.15
For x = 9, ŷ = 3.75 - 0.15 * 9 = 3.00
Now we can calculate the value of e (residual) for each point by subtracting y from ŷ:
For x = 1, e = 3 - 3.60 = -0.60
For x = 3, e = 2 - 3.45 = -1.45
For x = 5, e = 6 - 3.30 = 2.70
For x = 7, e = 1 - 3.15 = -2.15
For x = 9, e = 3 - 3.00 = 0.00
The variance se² of the points about the line of best fit can be calculated using the formula:
se² = Σe² / (n - 2)
where Σe² is the sum of the squares of the residuals and n is the number of data points.
Σe² = (-0.60)² + (-1.45)² + 2.70² + (-2.15)² + 0.00² = 15.445
se² = 15.445 / (10 - 2) = 1.932
Therefore, the variance se² of the points about the line of best fit is 1.932.