Math Problem Statement

Trouvez la droite de régression des moindres carrés \( \hat{y} = b_0 + b_1 x \) à travers les points \((-3, 0), (3, 9), (4, 14), (9, 18), (11, 25)\), et utilisez-la pour trouver les estimations ponctuelles \( \hat{y} \) correspondantes à \( x = 3 \) et \( x = 6 \).

Solution

To find the least squares regression line y^=b0+b1x\hat{y} = b_0 + b_1 x for the given points, we'll follow these steps:

  1. Calculate the Means: Find the mean of xx and yy values.
  2. Calculate b1b_1 (Slope): b1=(xxˉ)(yyˉ)(xxˉ)2b_1 = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2}
  3. Calculate b0b_0 (Intercept): b0=yˉb1xˉb_0 = \bar{y} - b_1 \bar{x}
  4. Formulate the Regression Line: Substitute b0b_0 and b1b_1 to get y^=b0+b1x\hat{y} = b_0 + b_1 x.
  5. Estimate y^\hat{y} for x=3x = 3 and x=6x = 6.

Let me calculate each step using the data points (3,0),(3,9),(4,14),(9,18),(11,25)(-3, 0), (3, 9), (4, 14), (9, 18), (11, 25).The least squares regression line is given by:

y^=5.13+1.68x\hat{y} = 5.13 + 1.68x

Using this line:

  • For x=3x = 3, y^10.17\hat{y} \approx 10.17
  • For x=6x = 6, y^15.22\hat{y} \approx 15.22

Would you like more details on any of these calculations or the interpretation?

  1. How would the regression change with different points?
  2. Why are these particular estimations useful?
  3. How does this regression line minimize error?
  4. What does the slope represent in this context?
  5. How does this method differ from other regression approaches?

Tip: Double-check calculations by substituting points back into the regression line for validation.

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Math Problem Analysis

Mathematical Concepts

Regression Analysis
Least Squares Method
Linear Equations

Formulas

Least Squares Regression Line: \( \hat{y} = b_0 + b_1 x \)
Slope (b1): \( b_1 = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2} \)
Intercept (b0): \( b_0 = \bar{y} - b_1 \bar{x} \)

Theorems

Least Squares Estimation

Suitable Grade Level

Grades 11-12