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IB Mathematics AA Equation of the regression line Study Notes

IB Mathematics AA Equation of the regression line Study Notes

IB Mathematics AA Equation of the regression line Study Notes

IB Mathematics AA Equation of the regression line Notes Offer a clear explanation of Equation of the regression line, its mean and variance, including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Equation of the regression line, its mean and variance.

Equation of the Regression Line

Introduction

Regression analysis is used to model the relationship between two variables and make predictions. The regression line of \( X \) on \( Y \) is a linear equation that best fits a given dataset.

Key Concepts

1. Equation of the Regression Line

  • The equation of the regression line of \( X \) on \( Y \) is given by:

    \( x = a + by \)

    where:

    • \( a \) is the intercept, representing the value of \( X \) when \( Y = 0 \).
    • \( b \) is the slope, indicating how \( X \) changes with \( Y \).
  • The slope \( b \) is computed using:

    \( b = \frac{\text{Cov}(X, Y)}{\text{Var}(Y)} \)

2. Use of the Regression Equation for Prediction

  • Once the regression line is determined, it can be used to estimate \( X \) for given values of \( Y \).
  • However, predictions are only reliable within the range of observed data.
  • Extrapolating beyond the observed data range can lead to inaccurate predictions.

Guidance, Clarification, and Syllabus Links

  • Students should understand how regression lines are derived and their practical applications.
  • Recognize the limitations of regression-based predictions, especially when applied to real-world scenarios.

Solved Example

Problem: A dataset provides the following values for \( X \) and \( Y \):

\( X \)\( Y \)
23
47
611

Find the regression line equation of \( X \) on \( Y \).

Step 1: Calculate Means of \( X \) and \( Y \)

\( \bar{X} = \frac{2 + 4 + 6}{3} = \frac{12}{3} = 4 \)

\( \bar{Y} = \frac{3 + 7 + 11}{3} = \frac{21}{3} = 7 \)

Step 2: Calculate Variance of \( Y \)

\( \text{Var}(Y) = \frac{(3-7)^2 + (7-7)^2 + (11-7)^2}{3} \)

\( = \frac{(-4)^2 + 0^2 + 4^2}{3} \)

\( = \frac{16 + 0 + 16}{3} = \frac{32}{3} \)

Step 3: Calculate Covariance of \( X \) and \( Y \)

\( \text{Cov}(X, Y) = \frac{(2-4)(3-7) + (4-4)(7-7) + (6-4)(11-7)}{3} \)

\( = \frac{(-2)(-4) + (0)(0) + (2)(4)}{3} \)

\( = \frac{8 + 0 + 8}{3} = \frac{16}{3} \)

Step 4: Calculate Slope \( b \)

\( b = \frac{\text{Cov}(X, Y)}{\text{Var}(Y)} = \frac{\frac{16}{3}}{\frac{32}{3}} = \frac{16}{32} = 0.5 \)

Step 5: Calculate Intercept \( a \)

\( a = \bar{X} – b\bar{Y} = 4 – (0.5 \times 7) \)

\( = 4 – 3.5 = 0.5 \)

Step 6: Write the Regression Equation

\( x = 0.5 + 0.5y \)

IB Mathematics AA SL Equation of the regression line Exam Style Worked Out Questions

[MAI 4.4] LINEAR REGRESSION-manav-ready

Question

[Maximum mark: 7]
Consider the following data

(a) Find the correlation coefficient r. [1]
(b) Describe the relation between x and y. [2]
(c) Find the equation y = ax+b of the regression line for y on x. [2]
(d) Describe what the coefficient a represents. [1]
(e) Describe what the constant b represents. [1]

▶️Answer/Explanation

Ans:
(a) 0.965
(b) strong positive
(c) y = 2.2x – 0.5
(d) whenever x increases by 1 unit, y increases by 2.2 units.
(e) The value of y corresponding to 0 units of x.

Question

[Maximum mark: 6]
Consider the following data

The regression line for y on x is y = 2.2x – 0.5
(a) Solve the equation above for x to find an expression in the form x = ay+b [2]
(b) Find the equation x = cy+d of the regression line for x on y. [2]
(c) Describe the advantage of the linear equation in (b). [2]

▶️Answer/Explanation

Ans:
(a) y = 2.2x – 0.5 ⇔ y + 0.5 = 2.2x ⇔ x = 0.455 y + 0.227
(b) x = 0.423y + 0.385
(c) The relation in (a) is in fact the inverse function of the line y = 2.2x – 0.5
If y is given, the answer in (c) gives a more reliable estimation of x.

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