If the regression equation of two variables are 5x - y = 4 and 3x - 2y = 1 . Find the arithmetic means of x and y

Option c: 1, 1. Solution: We know that the two regression lines always intersect at the point of their arithmetic means, (, ) . Therefore, to find the arithmetic means, we solve the two regression equations simultaneously: 1) 5 - = 4 2) 3 - 2 = 1 From equation (1), we can express in terms of : = 5 - 4 Substitute this expression for into equation (2): 3 - 2(5 - 4) = 1 3 - 10 + 8 = 1 -7 + 8 = 1 -7 = 1 - 8 -7 = -7 = 1 Now, substitute = 1 back into the expression for : = 5(1) - 4 = 1 Thus, the arithmetic means are = 1 and = 1 , representing the pair (1, 1) . Hence, Option C is the correct answer.

If the regression equation of two variables are $5x - y = 4$ and $3x - 2y = 1$. Find the arithmetic means of $x$ and $y$

The correct answer is Option c: $1, 1$. We know that the two regression lines always intersect at the point of their arithmetic means, $(\\bar{x}, \\bar{y})$. Therefore, to find the arithmetic means, we solve the two regression equations simultaneously: 1) $5\\bar{x} - \\bar{y} = 4$ 2) $3\\bar{x} - 2\\bar{y} = 1$ From equation (1), we can express $\\bar{y}$ in terms of $\\bar{x}$: $\\bar{y} = 5\\bar{x} - 4$ Substitute this expression for $\\bar{y}$ into equation (2): $3\\bar{x} - 2(5\\bar{x} - 4) = 1$ $3\\bar{x} - 10\\bar{x} + 8 = 1$ $-7\\bar{x} + 8 = 1$ $-7\\bar{x} = 1 - 8$ $-7\\bar{x} = -7$ $\\bar{x} = 1$ Now, substitute $\\bar{x} = 1$ back into the expression for $\\bar{y}$: $\\bar{y} = 5(1) - 4 = 1$ Thus, the arithmetic means are $\\bar{x} = 1$ and $\\bar{y} = 1$, representing the pair $(1, 1)$. Hence, **Option C** is the correct answer.

Correlation and RegressionMCQMTP Dec 22 Series IQuestion 3765 of 188

All Questions

If the regression equation of two variables are $\displaystyle 5x - y = 4$ and $\displaystyle 3x - 2y = 1$ . Find the arithmetic means of $\displaystyle x$ and $\displaystyle y$

Options

\displaystyle 2, 1

\displaystyle 2, 2

\displaystyle 1, 1

DCannot be determined

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

Correct Answer

✅ Option c — $\displaystyle 1, 1$

All Options:

A $\displaystyle 2, 1$
B $\displaystyle 2, 2$
C $\displaystyle 1, 1$
DCannot be determined

Detailed Solution & Explanation

We know that the two regression lines always intersect at the point of their arithmetic means,

\displaystyle (\\bar{x}, \\bar{y})

. Therefore, to find the arithmetic means, we solve the two regression equations simultaneously: 1)

\displaystyle 5\\bar{x} - \\bar{y} = 4

\displaystyle 3\\bar{x} - 2\\bar{y} = 1

From equation (1), we can express

\displaystyle \\bar{y}

in terms of

\displaystyle \\bar{x}

\displaystyle \\bar{y} = 5\\bar{x} - 4

Substitute this expression for

\displaystyle \\bar{y}

into equation (2):

\displaystyle 3\\bar{x} - 2(5\\bar{x} - 4) = 1

\displaystyle 3\\bar{x} - 10\\bar{x} + 8 = 1

\displaystyle -7\\bar{x} + 8 = 1

\displaystyle -7\\bar{x} = 1 - 8

\displaystyle -7\\bar{x} = -7

\displaystyle \\bar{x} = 1

Now, substitute

\displaystyle \\bar{x} = 1

back into the expression for

\displaystyle \\bar{y}

\displaystyle \\bar{y} = 5(1) - 4 = 1

Thus, the arithmetic means are

\displaystyle \\bar{x} = 1

and

\displaystyle \\bar{y} = 1

, representing the pair

\displaystyle (1, 1)

. Hence, **Option C** 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.

Related Comparison Tables

Correlation vs Regression: Statistics for CA Foundation

Ready to Master Correlation and Regression?

Practice all 188 questions with instant feedback, earn XP, track your streaks, and ace your CA Foundation exam.

Start Practicing — It's Free

If the regression equation of two variables are 5x−y=4\displaystyle 5x - y = 45x−y=4 and 3x−2y=1\displaystyle 3x - 2y = 13x−2y=1. Find the arithmetic means of x\displaystyle xx and y\displaystyle yy