The equations of the two lines of regression are 4x + 3y + 7 = 0 and 3x + 4y + 8 = 0 . Find the correlation coefficient between x and y .

Option a: -0.75. Solution: We are given the two regression equations:1. 4x + 3y + 7 = 0 2. 3x + 4y + 8 = 0 us assume equation (1) is the regression line of y on x and equation (2) is the regression line of x on y : (1): 3y = -4x - 7 y = -4/3x - 7/3 byx = -4/3 ≈ -1.33 (2): 3x = -4y - 8 x = -4/3y - 8/3 bxy = -4/3 ≈ -1.33 the coefficient of determination r2 : r2 = byx × bxy = (-4/3) × (-4/3) = 16/9 ≈ 1.78 r2 must lie in the range [0, 1] , this assumption is invalid. Thus, we reverse our assumption. equation (2) be the regression line of y on x and equation (1) be the regression line of x on y : (2): 4y = -3x - 8 y = -0.75x - 2 byx = -0.75 (1): 4x = -3y - 7 x = -0.75y - 1.75 bxy = -0.75 , calculating r2 : r2 = byx × bxy = (-0.75) × (-0.75) = 0.5625 0 ≤ r2 ≤ 1 , this is a valid assumption. The correlation coefficient r must have the same sign as the regression coefficients, which is negative: r = -(r2) = -(0.5625) = -0.75 : The correct option in the source database is listed as Option B (1.25), which is impossible because correlation coefficients must lie between -1 and 1 . The mathematically correct option is Option A ( -0.75 )., Option A is the correct answer.

The equations of the two lines of regression are $4x + 3y + 7 = 0$ and $3x + 4y + 8 = 0$. Find the correlation coefficient between $x$ and $y$.

The correct answer is Option a: $-0.75$. We are given the two regression equations:\n1. $4x + 3y + 7 = 0$\n2. $3x + 4y + 8 = 0$\n\nLet us assume equation (1) is the regression line of $y$ on $x$ and equation (2) is the regression line of $x$ on $y$:\nFrom (1): $3y = -4x - 7 \\implies y = -\\frac{4}{3}x - \\frac{7}{3} \\implies b_{yx} = -\\frac{4}{3} \\approx -1.33$\nFrom (2): $3x = -4y - 8 \\implies x = -\\frac{4}{3}y - \\frac{8}{3} \\implies b_{xy} = -\\frac{4}{3} \\approx -1.33$\n\nCalculating the coefficient of determination $r^2$:\n$$r^2 = b_{yx} \\times b_{xy} = \\left(-\\frac{4}{3}\\right) \\times \\left(-\\frac{4}{3}\\right) = \\frac{16}{9} \\approx 1.78$$\nSince $r^2$ must lie in the range $[0, 1]$, this assumption is invalid. Thus, we reverse our assumption.\n\nLet equation (2) be the regression line of $y$ on $x$ and equation (1) be the regression line of $x$ on $y$:\nFrom (2): $4y = -3x - 8 \\implies y = -0.75x - 2 \\implies b_{yx} = -0.75$\nFrom (1): $4x = -3y - 7 \\implies x = -0.75y - 1.75 \\implies b_{xy} = -0.75$\n\nNow, calculating $r^2$:\n$$r^2 = b_{yx} \\times b_{xy} = (-0.75) \\times (-0.75) = 0.5625$$\nSince $0 \\le r^2 \\le 1$, this is a valid assumption. The correlation coefficient $r$ must have the same sign as the regression coefficients, which is negative:\n$$r = -\\sqrt{r^2} = -\\sqrt{0.5625} = -0.75$$\n\nNote: The correct option in the source database is listed as Option B (1.25), which is impossible because correlation coefficients must lie between $-1$ and $1$. The mathematically correct option is Option A ($-0.75$).\n\nHence, **Option A** is the correct answer.

Correlation and RegressionMCQMTP Dec 2023 Series IQuestion 3759 of 188

All Questions

The equations of the two lines of regression are $\displaystyle 4x + 3y + 7 = 0$ and $\displaystyle 3x + 4y + 8 = 0$ . Find the correlation coefficient between $\displaystyle x$ and $\displaystyle y$ .

Options

\displaystyle -0.75

\displaystyle 1.25

\displaystyle -0.92

\displaystyle 0.25

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Correct Answer

✅ Option a — $\displaystyle -0.75$

All Options:

A $\displaystyle -0.75$
B $\displaystyle 1.25$
C $\displaystyle -0.92$
D $\displaystyle 0.25$

Detailed Solution & Explanation

We are given the two regression equations:\n1.

\displaystyle 4x + 3y + 7 = 0

\n2.

\displaystyle 3x + 4y + 8 = 0

\n\nLet us assume equation (1) is the regression line of

\displaystyle y

\displaystyle x

and equation (2) is the regression line of

\displaystyle x

\displaystyle y

:\nFrom (1):

\displaystyle 3y = -4x - 7 \\implies y = -\\frac{4}{3}x - \\frac{7}{3} \\implies b_{yx} = -\\frac{4}{3} \\approx -1.33

\nFrom (2):

\displaystyle 3x = -4y - 8 \\implies x = -\\frac{4}{3}y - \\frac{8}{3} \\implies b_{xy} = -\\frac{4}{3} \\approx -1.33

\n\nCalculating the coefficient of determination

\displaystyle r^2

:\n

r^2 = b_{yx} \\times b_{xy} = \\left(-\\frac{4}{3}\\right) \\times \\left(-\\frac{4}{3}\\right) = \\frac{16}{9} \\approx 1.78

\nSince

\displaystyle r^2

must lie in the range

\displaystyle [0, 1]

, this assumption is invalid. Thus, we reverse our assumption.\n\nLet equation (2) be the regression line of

\displaystyle y

\displaystyle x

and equation (1) be the regression line of

\displaystyle x

\displaystyle y

:\nFrom (2):

\displaystyle 4y = -3x - 8 \\implies y = -0.75x - 2 \\implies b_{yx} = -0.75

\nFrom (1):

\displaystyle 4x = -3y - 7 \\implies x = -0.75y - 1.75 \\implies b_{xy} = -0.75

\n\nNow, calculating

\displaystyle r^2

:\n

r^2 = b_{yx} \\times b_{xy} = (-0.75) \\times (-0.75) = 0.5625

\nSince

\displaystyle 0 \\le r^2 \\le 1

, this is a valid assumption. The correlation coefficient

\displaystyle r

must have the same sign as the regression coefficients, which is negative:\n

r = -\\sqrt{r^2} = -\\sqrt{0.5625} = -0.75

\n\nNote: The correct option in the source database is listed as Option B (1.25), which is impossible because correlation coefficients must lie between

\displaystyle -1

and

\displaystyle 1

. The mathematically correct option is Option A (

\displaystyle -0.75

).\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

Correlation

A statistical measure that expresses the extent to which two variables are linearly related.

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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The equations of the two lines of regression are 4x+3y+7=0\displaystyle 4x + 3y + 7 = 04x+3y+7=0 and 3x+4y+8=0\displaystyle 3x + 4y + 8 = 03x+4y+8=0. Find the correlation coefficient between x\displaystyle xx and y\displaystyle yy.