Math Problem Statement

Find the least-squares line y = β0 + β1x that best fits the data (-4,0), (-1,0), (0,3), (1,5), and (4,5), assuming that the first and last data points are less reliable. Weight them half as much as the three interior points.

Solution

To solve this problem, we need to calculate the weighted least-squares regression line y=β0+β1xy = \beta_0 + \beta_1 x. The weights will account for the different reliabilities of the data points.

Steps:

  1. Weight Assignment:

    • First and last points: weight = 0.5.
    • Interior points: weight = 1.
  2. Weighted Least-Squares Formulas:

    • Let the weights be wiw_i, the x-values be xix_i, and the y-values be yiy_i.
    • The formulas for β1\beta_1 (slope) and β0\beta_0 (intercept) are: β1=wi(xixˉw)(yiyˉw)wi(xixˉw)2\beta_1 = \frac{\sum w_i (x_i - \bar{x}_w)(y_i - \bar{y}_w)}{\sum w_i (x_i - \bar{x}_w)^2} β0=yˉwβ1xˉw\beta_0 = \bar{y}_w - \beta_1 \bar{x}_w where xˉw=wixiwi\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i} and yˉw=wiyiwi\bar{y}_w = \frac{\sum w_i y_i}{\sum w_i}.
  3. Data: Points and weights:

    • (4,0),w=0.5(-4, 0), w = 0.5
    • (1,0),w=1(-1, 0), w = 1
    • (0,3),w=1(0, 3), w = 1
    • (1,5),w=1(1, 5), w = 1
    • (4,5),w=0.5(4, 5), w = 0.5
  4. Calculate Weighted Means:

    • xˉw\bar{x}_w and yˉw\bar{y}_w will be calculated using the weights.
  5. Apply the Weighted Least-Squares Formulas: Plug in the data, calculate β1\beta_1 and β0\beta_0.

Let me calculate this for you. numpy as np

Data points

x = np.array([-4, -1, 0, 1, 4]) y = np.array([0, 0, 3, 5, 5]) weights = np.array([0.5, 1, 1, 1, 0.5])

Weighted means

weighted_mean_x = np.sum(weights * x) / np.sum(weights) weighted_mean_y = np.sum(weights * y) / np.sum(weights)

Weighted least-squares slope (beta_1)

numerator = np.sum(weights * (x - weighted_mean_x) * (y - weighted_mean_y)) denominator = np.sum(weights * (x - weighted_mean_x) ** 2) beta_1 = numerator / denominator

Intercept (beta_0)

beta_0 = weighted_mean_y - beta_1 * weighted_mean_x

beta_0, beta_1The weighted least-squares regression line is:

y=2.625+0.833xy = 2.625 + 0.833x

Would you like a detailed explanation of how the weights influenced the result, or assistance with a related question?

Related Questions:

  1. How does the weighting impact the regression line compared to an unweighted fit?
  2. Can you provide the unweighted least-squares solution for comparison?
  3. What are the implications of assigning different weights in a regression analysis?
  4. How would adding more data points change the regression line?
  5. What is the standard error of the regression line for this fit?

Tip:

In weighted least-squares regression, assigning appropriate weights can significantly improve the accuracy of your model when some data points are less reliable.

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Math Problem Analysis

Mathematical Concepts

Weighted least-squares regression
Linear algebra
Statistics

Formulas

Weighted mean: x̄_w = Σ(w_i * x_i) / Σw_i
Weighted slope: β1 = Σ(w_i * (x_i - x̄_w) * (y_i - ȳ_w)) / Σ(w_i * (x_i - x̄_w)^2)
Intercept: β0 = ȳ_w - β1 * x̄_w

Theorems

Weighted least-squares method

Suitable Grade Level

Undergraduate Statistics or High School Advanced Math