Home / IB DP Maths / IB Math Analysis and Approach HL / MAA HL Study Notes / Intersections of a line with a planes 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 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

The regression line represents the best-fit line through a set of data points in a scatterplot. It is used to describe the relationship between two variables and make predictions about one variable based on the value of another.

Key Concepts

1. Equation of the Regression Line
  • The regression line of \( X \) on \( Y \) is expressed as:

    \( x = a + by \)

    where:
    • \( a \): Intercept of the regression line on the x-axis.
    • \( b \): Slope of the regression line, representing the rate of change of \( X \) with respect to \( Y \).
  • Key features:
    • Regression lines minimize the sum of the squared vertical distances between observed points and the line.
    • The slope (\( b \)) is calculated as:

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

2. Prediction Using the Regression Line
  • The regression line allows for predictions of \( X \) based on \( Y \), but these predictions are only reliable within the range of observed data.
  • Extrapolation outside the observed data range can lead to unreliable or inaccurate predictions.

Guidance, Clarification, and Syllabus Links

  • Students should understand the practical applications of regression lines for prediction and analysis.
  • Be cautious about the limitations of prediction using regression, especially for values outside the observed range.

Connections and Extensions

1. Theory of Knowledge (TOK)
  • Discussion: Is it possible to have knowledge of the future? The reliability of regression-based predictions highlights the challenges of making assumptions about future outcomes based on historical data.
2. Enrichment
  • Explore multiple regression analysis involving more than two variables.
  • Study the role of correlation in understanding the strength and direction of relationships between variables.

Solved Example

Example 1: Finding the Regression Line

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

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

Calculate the regression line of \( X \) on \( Y \).

Solution:

  • Find the mean and variance of \( Y \), and the covariance of \( X \) and \( Y \).
  • Use the formulas:

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

    \( a = \bar{x} – b\bar{y} \)

Answer: The regression line equation is \( x = 1.5 + 0.75y \).

Conclusion

The regression line provides a valuable tool for understanding relationships between variables and making predictions. However, its limitations must be acknowledged, particularly regarding the reliability of predictions outside the observed data range.

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

Question

Let X be normally distributed with mean 100 cm and standard deviation 5 cm.

a.On the diagram below, shade the region representing \({\rm{P}}(X > 105)\) .[2]


 
 

b.Given that \({\rm{P}}(X < d) = {\rm{P}}(X > 105)\) , find the value of \(d\) . [2]

c.Given that \({\rm{P}}(X > 105) = 0.16\) (correct to two significant figures), find \({\rm{P}}(d < X < 105)\) . [2]

▶️Answer/Explanation

Markscheme

a.
     A1A1     N2

Note: Award A1 for vertical line to right of mean, A1 for shading to right of their vertical line.

b.

evidence of recognizing symmetry     (M1)

e.g. \(105\) is one standard deviation above the mean so \(d\) is one standard deviation below the mean, shading the corresponding part, \(105 – 100 = 100 – d\)

\(d = 95\)     A1     N2

[2 marks]

c.

evidence of using complement     (M1)

e.g. \(1 – 0.32\) , \(1 – p\)

\({\rm{P}}(d < X < 105) = 0.68\)     A1     N2

[2 marks]

 

Question

The random variable X is normally distributed with mean 1000 and standard deviation 50.

(a) Find   (i) \(P(X < 925)\)    (ii) \(P(925 < X < 1025)\)    (iii) \(P(X > 1025)\)

(b) Sketch a graph representing the information in \((a)\)

(c) Find the standardised values of \(925\) and \(1025\)

(d) Sketch the corresponding graph of standardised values

(e) Find \(E(X^{2})\)

▶️Answer/Explanation

Ans

(a)    (i) \(P(X<925)=0.0668\)              (ii)  \(P(925<X<1025)\)                (iii)  \(P(X>1025)=0.309\)

(b)                 

(c)    \(-1.5\) and \(0.5\) respectively

(d)    similar to the above the vertical boundaries are \(-1.5\) and \(0.5\)

(e)    \(E(X^{2})=Var(X)+E(X)^{2}=\sigma ^{2}+\mu ^{2}=50^{2}+1000^{2}=1002500\)

More resources for IB Mathematics AA SL

Scroll to Top