Correlation and RegressionMCQMTP Nov 18Question 3732 of 188
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For two variables x\displaystyle x and y\displaystyle y with the same mean the regression equation are y=2x−α\displaystyle y = 2x - \alpha and x=2y−β\displaystyle x = 2y - \beta, what is the value of common mean

Options

A−α\displaystyle -\alpha
Bβ\displaystyle \beta
C0\displaystyle 0
D−β\displaystyle -\beta
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Correct Answer

✅ Option b — β\displaystyle \beta

All Options:

  • A−α\displaystyle -\alpha
  • Bβ\displaystyle \beta
  • C0\displaystyle 0
  • D−β\displaystyle -\beta

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Detailed Solution & Explanation

Let the common mean of x\displaystyle x and y\displaystyle y be denoted by mu\displaystyle \\mu, so that barx=bary=mu\displaystyle \\bar{x} = \\bar{y} = \\mu.\n\nSince regression lines always pass through the mean coordinates (barx,bary)\displaystyle (\\bar{x}, \\bar{y}), we can substitute these mean values into the given regression equations:\n\n1. For y=2x−alpha\displaystyle y = 2x - \\alpha:\nmu=2mu−alphaimpliesmu=alpha\\mu = 2\\mu - \\alpha \\implies \\mu = \\alpha\n\n2. For x=2y−beta\displaystyle x = 2y - \\beta:\nmu=2mu−betaimpliesmu=beta\\mu = 2\\mu - \\beta \\implies \\mu = \\beta\n\nFrom these relations, we obtain:\nmu=alpha=beta\\mu = \\alpha = \\beta\n\nThus, the common mean mu\displaystyle \\mu is equal to beta\displaystyle \\beta (or alpha\displaystyle \\alpha). Comparing this with the choices:\n- Option A: −alpha\displaystyle -\\alpha\n- Option B: beta\displaystyle \\beta\n- Option C: 0\displaystyle 0\n- Option D: −beta\displaystyle -\\beta\n\nWe see that mu=beta\displaystyle \\mu = \\beta corresponds to Option B. The source key's correct_option of 'a' (−alpha\displaystyle -\\alpha) is incorrect. The correct option is B.\n\nHence, **Option B** is the correct answer.

About This Chapter: Correlation and Regression

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Correlation Coefficient, Regression Equations

This chapter covers Correlation Coefficient, Regression Equations and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 4-6 Marks weightage. Focus on understanding core concepts rather than memorizing.

Key Concepts to Understand

Regression

A statistical method used for estimating the relationships between a dependent variable and one or more independent variables.

Related Comparison Tables

Correlation vs Regression: Statistics for CA Foundation

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