Correlation and RegressionMCQMTP Dec 2023 Series IQuestion 3759 of 188
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The equations of the two lines of regression are $4x + 3y + 7 = 0\displaystyle and $3x + 4y + 8 = 0. Find the correlation coefficient between x\displaystyle x and y\displaystyle y.

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option a — −0.75\displaystyle -0.75

All Options:

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

Detailed Solution & Explanation

We are given the two regression equations:\n1. $4x + 3y + 7 = 0\displaystyle \n2. $3x + 4y + 8 = 0\n\nLet us assume equation (1) is the regression line of y\displaystyle y on x\displaystyle x and equation (2) is the regression line of x\displaystyle x on y\displaystyle y:\nFrom (1): $3y = -4x - 7 \\implies y = -\43\displaystyle \frac{4}{3}x - \73\displaystyle \frac{7}{3} \\implies b_{yx} = -\43\displaystyle \frac{4}{3} \≈−1\displaystyle \approx -1.33\displaystyle \nFrom (2): $3x = -4y - 8 \\implies x = -\43\displaystyle \frac{4}{3}y - \83\displaystyle \frac{8}{3} \\implies b_{xy} = -\43\displaystyle \frac{4}{3} \≈−1\displaystyle \approx -1.33\n\nCalculating the coefficient of determination r2\displaystyle r^2:\nr2=byxtimesbxy=left(−frac43right)timesleft(−frac43right)=frac169approx1.78r^2 = b_{yx} \×\displaystyle \times b_{xy} = \\left(-\43\displaystyle \frac{4}{3}\\right) \×\displaystyle \times \\left(-\43\displaystyle \frac{4}{3}\\right) = \169\displaystyle \frac{16}{9} \≈\displaystyle \approx 1.78\nSince r2\displaystyle r^2 must lie in the range [0,1]\displaystyle [0, 1], this assumption is invalid. Thus, we reverse our assumption.\n\nLet equation (2) be the regression line of y\displaystyle y on x\displaystyle x and equation (1) be the regression line of x\displaystyle x on y\displaystyle y:\nFrom (2): $4y = -3x - 8 \\implies y = -0.75x - 2 \\implies b_{yx} = -0.75\displaystyle \nFrom (1): $4x = -3y - 7 \\implies x = -0.75y - 1.75 \\implies b_{xy} = -0.75\n\nNow, calculating r2\displaystyle r^2:\nr2=byxtimesbxy=(−0.75)times(−0.75)=0.5625r^2 = b_{yx} \×\displaystyle \times b_{xy} = (-0.75) \×\displaystyle \times (-0.75) = 0.5625\nSince 0ler2le1\displaystyle 0 \≤\displaystyle \le r^2 \≤\displaystyle \le 1, this is a valid assumption. The correlation coefficient r\displaystyle r must have the same sign as the regression coefficients, which is negative:\nr=−sqrtr2=−sqrt0.5625=−0.75r = -\r2\displaystyle \sqrt{r^2} = -\0.5625\displaystyle \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\displaystyle -1 and 1\displaystyle 1. The mathematically correct option is Option A (−0.75\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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