Correlation and RegressionMCQMTP Oct 21Question 3750 of 188
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If the regression line of y\displaystyle y on x\displaystyle x and of x\displaystyle x on y\displaystyle y are given by 2x+3y=1\displaystyle 2x + 3y = -1 and 5x+6y=1\displaystyle 5x + 6y = -1, then the arithmetic means of x\displaystyle x and y\displaystyle y are given by

Options

A(1,1)\displaystyle (1, -1)
B(2,3)\displaystyle (2, 3)
C(1,1)\displaystyle (-1, -1)
D(2,3)\displaystyle (2, -3)
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Correct Answer

Option a(1,1)\displaystyle (1, -1)

All Options:

  • A(1,1)\displaystyle (1, -1)
  • B(2,3)\displaystyle (2, 3)
  • C(1,1)\displaystyle (-1, -1)
  • D(2,3)\displaystyle (2, -3)

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Detailed Solution & Explanation

The intersection point of the two regression lines is the point of their arithmetic means (barx,bary)\displaystyle (\\bar{x}, \\bar{y}). Therefore, we solve the two given equations simultaneously:\n1) 2x+3y=1\displaystyle 2x + 3y = -1\n2) 5x+6y=1\displaystyle 5x + 6y = -1\n\nMultiply equation (1) by 2\displaystyle 2:\n4x+6y=24x + 6y = -2\n\nSubtract this from equation (2):\n(5x+6y)(4x+6y)=1(2)(5x + 6y) - (4x + 6y) = -1 - (-2)\nx=1x = 1\n\nSubstitute x=1\displaystyle x = 1 back into equation (1):\n\displaystyle &#x27; in math mode at position 15: 2(1) + 3y = -1̲\n" style="color:#cc0000">2(1) + 3y = -1\n</span>3y=3impliesy=1\displaystyle \n</span>3y = -3 \\implies y = -1\n\nThus,thearithmeticmeansof\displaystyle \n\nThus, the arithmetic means ofxand\displaystyle andyare\displaystyle are(\\bar{x}, \\bar{y}) = (1, -1)$, which corresponds to Option A. (Note: The source file indicating Option D is a typo).\n\nHence, **Option A** 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

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