Correlation and RegressionMCQMTP Dec 22 Series IIQuestion 3751 of 188
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If the two regression lines are $3X = Y\displaystyle and $8Y = 6X then the value of correlation coefficient is:

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

✅ Option b — $0.5$

All Options:

  • A−0.5\displaystyle -0.5
  • B$0.5$
  • C$0.75$
  • D−0.80\displaystyle -0.80

Detailed Solution & Explanation

We are given the two regression lines:\n1) $3X = Y \\implies X = \13\displaystyle \frac{1}{3}Y\nIfweassumethisistheregressionlineof\displaystyle \nIf we assume this is the regression line ofXon\displaystyle onY\displaystyle , the regression coefficient is:\n<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>b</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>=</mo><mspace linebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>13</mn></mrow><annotation encoding="application/x-tex">b_{xy} = \13\displaystyle \frac{1}{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span><span class="mord"><span class="mord">1</span></span><span class="mord"><span class="mord">3</span></span></span></span></span></span>\n\n2) &#36;8Y = 6X \\implies Y = \68\displaystyle \frac{6}{8}X = \34\displaystyle \frac{3}{4}X\nIf we assume this is the regression line of Y\displaystyle Y on X\displaystyle X, the regression coefficient is:\nbyx=frac34b_{yx} = \34\displaystyle \frac{3}{4}\n\nThe coefficient of determination is:\nr2=bxycdotbyx=frac13cdotfrac34=frac14=0.25r^2 = b_{xy} \⋅\displaystyle \cdot b_{yx} = \13\displaystyle \frac{1}{3} \⋅\displaystyle \cdot \34\displaystyle \frac{3}{4} = \14=0\displaystyle \frac{1}{4} = 0.25\n\nSince 0ler2le1\displaystyle 0 \≤\displaystyle \le r^2 \≤\displaystyle \le 1, our assumptions are valid. The correlation coefficient is the geometric mean of the regression coefficients. Since both bxy\displaystyle b_{xy} and byx\displaystyle b_{yx} are positive, r\displaystyle r must be positive:\nr=sqrt0.25=0.5r = \0.25=0\displaystyle \sqrt{0.25} = 0.5\n\nNote: The source file indicating Option D is a typo.\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

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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