If two lines of regression are x+2y-5=0 and 2x+3y-8=0 , then the regression line of y on x is:

Option a: x+2y-5=0. Solution: To determine which equation represents the regression line of y on x , we make an assumption and verify it using the properties of regression coefficients. Assumption: Let the first line x + 2y - 5 = 0 be the regression line of y on x , and the second line 2x + 3y - 8 = 0 be the regression line of x on y . Step 1: Find the regression coefficient byx from the line of y on x Expressing x + 2y - 5 = 0 in terms of y : 2y = -x + 5 ⟹ y = -0.5x + 2.5 Thus, the slope is the regression coefficient of y on x : byx = -0.5 Step 2: Find the regression coefficient bxy from the line of x on y Expressing 2x + 3y - 8 = 0 in terms of x : 2x = -3y + 8 ⟹ x = -1.5y + 4 Thus, the slope is the regression coefficient of x on y : bxy = -1.5 Step 3: Verify the correlation coefficient property For the assumption to be valid, the product of the regression coefficients ( byx × bxy = r2 ) must be positive (since both coefficients have the same sign) and must satisfy 0 ≤ r2 ≤ 1 : r2 = byx × bxy = (-0.5) × (-1.5) = 0.75 Since 0 ≤ 0.75 ≤ 1 , this is a mathematically valid scenario. Thus, our initial assumption is correct. The regression line of y on x is indeed x + 2y - 5 = 0 . Hence, Option A is the correct answer.

If two lines of regression are $x+2y-5=0$ and $2x+3y-8=0$, then the regression line of $y$ on $x$ is:

The correct answer is Option a: $x+2y-5=0$. To determine which equation represents the regression line of $y$ on $x$, we make an assumption and verify it using the properties of regression coefficients. **Assumption:** Let the first line $x + 2y - 5 = 0$ be the regression line of $y$ on $x$, and the second line $2x + 3y - 8 = 0$ be the regression line of $x$ on $y$. **Step 1: Find the regression coefficient $b_{yx}$ from the line of $y$ on $x$** Expressing $x + 2y - 5 = 0$ in terms of $y$: $$2y = -x + 5 \Rightarrow y = -0.5x + 2.5$$ Thus, the slope is the regression coefficient of $y$ on $x$: $$b_{yx} = -0.5$$ **Step 2: Find the regression coefficient $b_{xy}$ from the line of $x$ on $y$** Expressing $2x + 3y - 8 = 0$ in terms of $x$: $$2x = -3y + 8 \Rightarrow x = -1.5y + 4$$ Thus, the slope is the regression coefficient of $x$ on $y$: $$b_{xy} = -1.5$$ **Step 3: Verify the correlation coefficient property** For the assumption to be valid, the product of the regression coefficients ($b_{yx} \times b_{xy} = r^2$) must be positive (since both coefficients have the same sign) and must satisfy $0 \le r^2 \le 1$: $$r^2 = b_{yx} \times b_{xy} = (-0.5) \times (-1.5) = 0.75$$ Since $0 \le 0.75 \le 1$, this is a mathematically valid scenario. Thus, our initial assumption is correct. The regression line of $y$ on $x$ is indeed $x + 2y - 5 = 0$. Hence, **Option A** is the correct answer.

Correlation and RegressionMCQPYQ Nov 18Question 3699 of 188

All Questions

If two lines of regression are $\displaystyle x+2y-5=0$ and $\displaystyle 2x+3y-8=0$ , then the regression line of $\displaystyle y$ on $\displaystyle x$ is:

Options

\displaystyle x+2y-5=0

\displaystyle 2x+3y-8=0

\displaystyle x+2y=0

\displaystyle 2x+3y=0

For any discrepancies in this question, email contact@cadada.in

Correct Answer

✅ Option a — $\displaystyle x+2y-5=0$

All Options:

A $\displaystyle x+2y-5=0$
B $\displaystyle 2x+3y-8=0$
C $\displaystyle x+2y=0$
D $\displaystyle 2x+3y=0$

Detailed Solution & Explanation

To determine which equation represents the regression line of

\displaystyle y

\displaystyle x

, we make an assumption and verify it using the properties of regression coefficients. **Assumption:** Let the first line

\displaystyle x + 2y - 5 = 0

be the regression line of

\displaystyle y

\displaystyle x

, and the second line

\displaystyle 2x + 3y - 8 = 0

be the regression line of

\displaystyle x

\displaystyle y

. **Step 1: Find the regression coefficient

\displaystyle b_{yx}

from the line of

\displaystyle y

\displaystyle x

** Expressing

\displaystyle x + 2y - 5 = 0

in terms of

\displaystyle y

2y = -x + 5 \Rightarrow y = -0.5x + 2.5

Thus, the slope is the regression coefficient of

\displaystyle y

\displaystyle x

b_{yx} = -0.5

**Step 2: Find the regression coefficient

\displaystyle b_{xy}

from the line of

\displaystyle x

\displaystyle y

** Expressing

\displaystyle 2x + 3y - 8 = 0

in terms of

\displaystyle x

2x = -3y + 8 \Rightarrow x = -1.5y + 4

Thus, the slope is the regression coefficient of

\displaystyle x

\displaystyle y

b_{xy} = -1.5

**Step 3: Verify the correlation coefficient property** For the assumption to be valid, the product of the regression coefficients (

\displaystyle b_{yx} \times b_{xy} = r^2

) must be positive (since both coefficients have the same sign) and must satisfy

\displaystyle 0 \le r^2 \le 1

r^2 = b_{yx} \times b_{xy} = (-0.5) \times (-1.5) = 0.75

Since

\displaystyle 0 \le 0.75 \le 1

, this is a mathematically valid scenario. Thus, our initial assumption is correct. The regression line of

\displaystyle y

\displaystyle x

is indeed

\displaystyle x + 2y - 5 = 0

. Hence, **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

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 two lines of regression are x+2y−5=0\displaystyle x+2y-5=0x+2y−5=0 and 2x+3y−8=0\displaystyle 2x+3y-8=02x+3y−8=0, then the regression line of y\displaystyle yy on x\displaystyle xx is: