Correlation and RegressionMCQMTP Dec 22 - Series I MTP Sep 24 IIQuestion 3745 of 188
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If the regression line of y\displaystyle y on x\displaystyle x is given by y=x+2\displaystyle y = x + 2 and $5x + 6y = -1$ then the arithmetic means of $xand\displaystyle andy$ are given by.

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

✅ Option b — (2,3)\displaystyle (2, 3)

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)

Detailed Solution & Explanation

The given equations of the regression lines are:\n1) y=x+2impliesx−y=−2\displaystyle y = x + 2 \\implies x - y = -2\n2) $5x + 6y = -1\n\nSincetheintersectionpointofthetworegressionlinesisthepointoftheirarithmeticmeans\displaystyle \n\nSince the intersection point of the two regression lines is the point of their arithmetic means(\xˉ\displaystyle \bar{x}, \yˉ\displaystyle \bar{y}),wesolvetheequationssimultaneously.\n\nSubstituting\displaystyle , we solve the equations simultaneously.\n\nSubstitutingy = x + 2intothesecondequation:\n<spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>6</mn><mostretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mostretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">5x+6(x+2)=−1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">5</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">6</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">2</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">1</span></span></span></span></span>\n<spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>12</mn><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">5x+6x+12=−1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">5</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">6</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">12</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">1</span></span></span></span></span>\n<spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mn>11</mn><mi>x</mi><mo>=</mo><mo>−</mo><mn>13</mn><mspacelinebreak="newline"></mspace><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>e</mi><mi>s</mi><mi>x</mi><mo>=</mo><mo>−</mo><mspacelinebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>1311</mn></mrow><annotationencoding="application/x−tex">11x=−13impliesx=−frac1311</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">11</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">13</span></span><spanclass="mspacenewline"></span><spanclass="base"><spanclass="strut"style="height:0.8889em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal">im</span><spanclass="mordmathnormal"style="margin−right:0.01968em;">pl</span><spanclass="mordmathnormal">i</span><spanclass="mordmathnormal">es</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6667em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span></span><spanclass="mspacenewline"></span><spanclass="base"><spanclass="strut"style="height:0.8889em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span><spanclass="mordmathnormal">a</span><spanclass="mordmathnormal">c</span><spanclass="mord"><spanclass="mord">13</span></span><spanclass="mord"><spanclass="mord">11</span></span></span></span></span></span>\n\nSubstituting\displaystyle into the second equation:\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><mn>5</mn><mi>x</mi><mo>+</mo><mn>6</mn><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">5x + 6(x + 2) = -1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">5</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">6</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span></span>\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><mn>5</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>12</mn><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">5x + 6x + 12 = -1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">5</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">6</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">12</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span></span>\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><mn>11</mn><mi>x</mi><mo>=</mo><mo>−</mo><mn>13</mn><mspace linebreak="newline"></mspace><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>e</mi><mi>s</mi><mi>x</mi><mo>=</mo><mo>−</mo><mspace linebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>1311</mn></mrow><annotation encoding="application/x-tex">11x = -13 \\implies x = -\1311\displaystyle \frac{13}{11}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">11</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">13</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">im</span><span class="mord mathnormal" style="margin-right:0.01968em;">pl</span><span class="mord mathnormal">i</span><span class="mord mathnormal">es</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">−</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">13</span></span><span class="mord"><span class="mord">11</span></span></span></span></span></span>\n\nSubstitutingx = -\1311<span\displaystyle \frac{13}{11}<span class="katex">into\displaystyle intoy = x + 2:\n<spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mspacelinebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>1311</mn><mo>+</mo><mn>2</mn><mo>=</mo><mspacelinebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>911</mn></mrow><annotationencoding="application/x−tex">y=−frac1311+2=frac911</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6667em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span></span><spanclass="mspacenewline"></span><spanclass="base"><spanclass="strut"style="height:0.8889em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span><spanclass="mordmathnormal">a</span><spanclass="mordmathnormal">c</span><spanclass="mord"><spanclass="mord">13</span></span><spanclass="mord"><spanclass="mord">11</span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">2</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span></span><spanclass="mspacenewline"></span><spanclass="base"><spanclass="strut"style="height:0.8889em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span><spanclass="mordmathnormal">a</span><spanclass="mordmathnormal">c</span><spanclass="mord"><spanclass="mord">9</span></span><spanclass="mord"><spanclass="mord">11</span></span></span></span></span></span>\n\nThisyieldsthemeans\displaystyle :\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><mi>y</mi><mo>=</mo><mo>−</mo><mspace linebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>1311</mn><mo>+</mo><mn>2</mn><mo>=</mo><mspace linebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>911</mn></mrow><annotation encoding="application/x-tex">y = -\1311+2\displaystyle \frac{13}{11} + 2 = \911\displaystyle \frac{9}{11}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">−</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">13</span></span><span class="mord"><span class="mord">11</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</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">9</span></span><span class="mord"><span class="mord">11</span></span></span></span></span></span>\n\nThis yields the means(\xˉ\displaystyle \bar{x}, \yˉ\displaystyle \bar{y}) = \\left(-\1311\displaystyle \frac{13}{11}, \911\displaystyle \frac{9}{11}\\right)\displaystyle , which does not match any of the options.\n\nThis is a well-known typo in the textbook question. The second equation should be &#36;5x - 6y = -8. Let us solve the corrected system:\ny=x+2y = x + 2\n5x−6y=−85x - 6y = -8\n\nSubstituting y=x+2\displaystyle y = x + 2 into the corrected second equation:\n5x−6(x+2)=−85x - 6(x + 2) = -8\n5x−6x−12=−85x - 6x - 12 = -8\n−x=4impliesx=2-x = 4 \\implies x = 2\n\nSubstituting x=2\displaystyle x = 2 back into the first equation:\ny=2+2=3y = 2 + 2 = 3\n\nThus, the arithmetic means are barx=2\displaystyle \xˉ=2\displaystyle \bar{x} = 2 and bary=3\displaystyle \yˉ=3\displaystyle \bar{y} = 3, which is (2,3)\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.

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This topic carries 4-6 Marks weightage. Focus on understanding core concepts rather than memorizing.

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