Given the following equation x,y data pairs find the least squares equation for 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 will determine the equation of a line that best fits the given data points by minimizing the sum of the squared differences between the observed and predicted values.
Let's perform the calculations step by step:
1. Calculate the mean of x and y:
- x̄ = (3 + 4 + 6) / 3 = 4.333 (rounded to three decimal places)
- ȳ = (3 + 5 + 8) / 3 = 5.333 (rounded to three decimal places)
2. Calculate the deviations from the mean for both x and y:
- Δx = [3 - 4.333, 4 - 4.333, 6 - 4.333] = [-1.333, -0.333, 1.667] (rounded to three decimal places)
- Δy = [3 - 5.333, 5 - 5.333, 8 - 5.333] = [-2.333, -0.333, 2.667] (rounded to three decimal places)
3. Calculate the sums of the products of the deviations from the mean:
- Σ(Δx * Δy) = (-1.333 * -2.333) + (-0.333 * -0.333) + (1.667 * 2.667) = 3.367 (rounded to three decimal places)
4. Calculate the sums of the squared deviations for x:
- Σ(Δx^2) = (-1.333)^2 + (-0.333)^2 + (1.667)^2 = 4.222 (rounded to three decimal places)
5. Calculate the slope (m) of the least squares line:
- m = Σ(Δx * Δy) / Σ(Δx^2) = 3.367 / 4.222 = 0.798 (rounded to three decimal places)
6. Calculate the y-intercept (b) of the least squares line:
- b = ȳ - (m * x̄) = 5.333 - (0.798 * 4.333) = 2.666 (rounded to three decimal places)
Therefore, the least squares equation for the given data is:
Y = 2.666 + 0.798x
Please note that the values are rounded to three decimal places as indicated.