given the following equations x,y data pairs find the least squares equation fo these data. use 3 decimal places. Y equals what + what x (3,3)(4,5)(6,8)
To find the least squares equation for the given data, we need to perform linear regression. Linear regression helps us find the best-fit line that minimizes the sum of the squared differences between the predicted and actual values.
Step 1: Find the mean (average) of x and y values:
- x̄ = (3 + 4 + 6) / 3 = 4.333 (to 3 decimal places)
- ȳ = (3 + 5 + 8) / 3 = 5.333 (to 3 decimal places)
Step 2: Calculate the differences between each x and y value and their respective means:
- Δx = (3 - 4.333), (4 - 4.333), (6 - 4.333) = -1.333, -0.333, 1.667 (to 3 decimal places)
- Δy = (3 - 5.333), (5 - 5.333), (8 - 5.333) = -2.333, -0.333, 2.667 (to 3 decimal places)
Step 3: Calculate the sum of the products of the differences:
- Σ(Δx * Δy) = (-1.333 * -2.333) + (-0.333 * -0.333) + (1.667 * 2.667) = 6.445 (to 3 decimal places)
Step 4: Calculate the sum of the squared differences of x:
- Σ(Δx^2) = (-1.333)^2 + (-0.333)^2 + (1.667)^2 = 4.110 (to 3 decimal places)
Step 5: Calculate the slope (b):
- b = Σ(Δx * Δy) / Σ(Δx^2) = 6.445 / 4.110 = 1.566 (to 3 decimal places)
Step 6: Calculate the y-intercept (a):
- a = ȳ - b * x̄ = 5.333 - (1.566 * 4.333) = -0.400 (to 3 decimal places)
Therefore, the least squares equation for the given data is: Y = -0.400 + 1.566x