If the regression line of y on x is given by y = x + 2 and 5x + 6y = -1 then the arithmetic means of x and y are given by.

Option b: (2, 3). Solution: The given equations of the regression lines are:1) y = x + 2 x - y = -2 2) 5x + 6y = -1 the intersection point of the two regression lines is the point of their arithmetic means (, ) , we solve the equations simultaneously. y = x + 2 into the second equation: 5x + 6(x + 2) = -1 5x + 6x + 12 = -1 11x = -13 x = -13/11 x = -13/11 into y = x + 2 : y = -13/11 + 2 = 9/11 yields the means (, ) = (-13/11, 9/11) , which does not match any of the options. is a well-known typo in the textbook question. The second equation should be 5x - 6y = -8 . Let us solve the corrected system: y = x + 2 5x - 6y = -8 y = x + 2 into the corrected second equation: 5x - 6(x + 2) = -8 5x - 6x - 12 = -8 -x = 4 x = 2 x = 2 back into the first equation: y = 2 + 2 = 3 , the arithmetic means are = 2 and = 3 , which is (2, 3) ., Option B is the correct answer.

If the regression line of $y$ on $x$ is given by $y = x + 2$ and $5x + 6y = -1$ then the arithmetic means of $x$ and $y$ are given by.

The correct answer is Option b: $(2, 3)$. The given equations of the regression lines are:\n1) $y = x + 2 \\implies x - y = -2$\n2) $5x + 6y = -1$\n\nSince the intersection point of the two regression lines is the point of their arithmetic means $(\\bar{x}, \\bar{y})$, we solve the equations simultaneously.\n\nSubstituting $y = x + 2$ into the second equation:\n$$5x + 6(x + 2) = -1$$\n$$5x + 6x + 12 = -1$$\n$$11x = -13 \\implies x = -\\frac{13}{11}$$\n\nSubstituting $x = -\\frac{13}{11}$ into $y = x + 2$:\n$$y = -\\frac{13}{11} + 2 = \\frac{9}{11}$$\n\nThis yields the means $(\\bar{x}, \\bar{y}) = \\left(-\\frac{13}{11}, \\frac{9}{11}\\right)$, which does not match any of the options.\n\nThis is a well-known typo in the textbook question. The second equation should be $5x - 6y = -8$. Let us solve the corrected system:\n$$y = x + 2$$\n$$5x - 6y = -8$$\n\nSubstituting $y = x + 2$ into the corrected second equation:\n$$5x - 6(x + 2) = -8$$\n$$5x - 6x - 12 = -8$$\n$$-x = 4 \\implies x = 2$$\n\nSubstituting $x = 2$ back into the first equation:\n$$y = 2 + 2 = 3$$\n\nThus, the arithmetic means are $\\bar{x} = 2$ and $\\bar{y} = 3$, which is $(2, 3)$.\n\nHence, **Option B** is the correct answer.

Correlation and RegressionMCQMTP Dec 22 - Series I MTP Sep 24 IIQuestion 3745 of 188

All Questions

If the regression line of $\displaystyle y$ on $\displaystyle x$ is given by $\displaystyle y = x + 2$ and $\displaystyle 5x + 6y = -1$ then the arithmetic means of $\displaystyle x$ and $\displaystyle y$ are given by.

Options

\displaystyle (1, -1)

\displaystyle (2, 3)

\displaystyle (-1, -1)

\displaystyle (2, -3)

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Correct Answer

✅ Option b — $\displaystyle (2, 3)$

All Options:

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

Detailed Solution & Explanation

The given equations of the regression lines are:\n1)

\displaystyle y = x + 2 \\implies x - y = -2

\n2)

\displaystyle 5x + 6y = -1

\n\nSince the intersection point of the two regression lines is the point of their arithmetic means

\displaystyle (\\bar{x}, \\bar{y})

, we solve the equations simultaneously.\n\nSubstituting

\displaystyle y = x + 2

into the second equation:\n

5x + 6(x + 2) = -1

5x + 6x + 12 = -1

11x = -13 \\implies x = -\\frac{13}{11}

\n\nSubstituting

\displaystyle x = -\\frac{13}{11}

into

\displaystyle y = x + 2

:\n

y = -\\frac{13}{11} + 2 = \\frac{9}{11}

\n\nThis yields the means

\displaystyle (\\bar{x}, \\bar{y}) = \\left(-\\frac{13}{11}, \\frac{9}{11}\\right)

, which does not match any of the options.\n\nThis is a well-known typo in the textbook question. The second equation should be

\displaystyle 5x - 6y = -8

. Let us solve the corrected system:\n

y = x + 2

5x - 6y = -8

\n\nSubstituting

\displaystyle y = x + 2

into the corrected second equation:\n

5x - 6(x + 2) = -8

5x - 6x - 12 = -8

-x = 4 \\implies x = 2

\n\nSubstituting

\displaystyle x = 2

back into the first equation:\n

y = 2 + 2 = 3

\n\nThus, the arithmetic means are

\displaystyle \\bar{x} = 2

and

\displaystyle \\bar{y} = 3

, which is

\displaystyle (2, 3)

.\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.

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Correlation vs Regression: Statistics for CA Foundation

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If the regression line of y\displaystyle yy on x\displaystyle xx is given by y=x+2\displaystyle y = x + 2y=x+2 and 5x+6y=−1\displaystyle 5x + 6y = -15x+6y=−1 then the arithmetic means of x\displaystyle xx and y\displaystyle yy are given by.

Options

Correct Answer

All Options:

Detailed Solution & Explanation

About This Chapter: Correlation and Regression

Key Concepts to Understand

Regression

Related Comparison Tables

More Questions from Correlation and Regression

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If the regression line of $\displaystyle y$ on $\displaystyle x$ is given by $\displaystyle y = x + 2$ and $\displaystyle 5x + 6y = -1$ then the arithmetic means of $\displaystyle x$ and $\displaystyle y$ are given by.