The information of investment (in lakh ₹) and its market price (In lakh ₹) after six months in share market in last seven years for a Mutual Fund Company is obtained as follows: | Particulars | Investment (Lakh ₹) $x$ | Market price after six months (Lakh ₹) $y$ | |---|---|---| | Mean | 40 | 50 | | Variance | 100 | 256 | Correlation Coefficient $= 0.5$ Obtain the regression line $Y$ on $X$ and estimate the market price in the share market after six months if there is an investment of ₹45 Lakh in a year. OR State properties of regression coefficient. Also state the point through which a regression line always passes.

The information of investment (in lakh ₹) and its market price (In lakh ₹) after six months in share market in

Question 27 of 42

The information of investment (in lakh ₹) and its market price (In lakh ₹) after six months in share market in last seven years for a Mutual Fund Company is obtained as follows:

Particulars	Investment (Lakh ₹) $x$	Market price after six months (Lakh ₹) $y$
Mean	40	50
Variance	100	256

Correlation Coefficient $= 0.5$

Obtain the regression line $Y$ on $X$ and estimate the market price in the share market after six months if there is an investment of ₹45 Lakh in a year.

State properties of regression coefficient. Also state the point through which a regression line always passes.

Gujarat GsebGujarat Board (GSEB) HSC Commerce Board 2023Subjective· 3mImportance★★★★★

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$S_x=10,S_y=16$ ; $b_{yx}=r\frac{S_y}{S_x}=0.5\times1.6=0.8$ ; $y-50=0.8(x-40)\Rightarrow y=0.8x+18$ ; $x=45\Rightarrow y=54$ . OR: properties of regression coefficients + line passes through $(\bar x,\bar y)$ .

Main part: Given $\bar x=40,\ \bar y=50$ , variance of $x=100\Rightarrow S_x=10$ , variance of $y=256\Rightarrow S_y=16$ , and $r=0.5$ .

$b_{yx}=r\cdot\frac{S_y}{S_x}=0.5\times\frac{16}{10}=0.5\times1.6=0.8.$

Regression line of $Y$ on $X$ through $(\bar x,\bar y)=(40,50)$ :

$y-50=0.8(x-40)\;\Rightarrow\;y=0.8x-32+50=0.8x+18.$

Estimate the market price for an investment of $x=45$ lakh:

$y=0.8(45)+18=36+18=54.$

So the estimated market price after six months is ₹54 lakh.

OR part — properties of regression coefficients:

Both regression coefficients $b_{yx}$ and $b_{xy}$ have the same sign (both positive or both negative), which is also the sign of $r$ .
The correlation coefficient is the geometric mean of the two: $r=\pm\sqrt{b_{yx}\cdot b_{xy}}$ . …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2026Set MARCH1 markMCQ
$Q. The best fitted line of regression can be obtained by which method?$
(a) Least Square Method
(b) Karl Pearson's Method
(c) Maximum Square Method
(d) Bowley's Method
›Reveal solutionSolution

The best-fitted regression line is found by the method of least squares.

The regression line is chosen so that the sum of the squares of the deviations of the observed $y$ from the estimated $\hat y$ is minimum, i.e. $\Sigma(y-\hat y)^2$ is least. This is the **least squa …

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CBSE 2026Set MARCH1 markMCQ
$\hat{y} = 30 - 1.5x$
(a) $28.5$
(b) $20$
(c) $15$
(d) $45$
›Reveal solutionSolution

The regression line passes through $(\bar x,\bar y)$ ; putting $\bar x=10$ gives $\bar y=15$ .

…

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CBSE 2025Set MARCH1 markMCQ
$Q. The regression line always passes through which point?$
(a) $(\bar{x}, \bar{y})$
(b) $(0, \bar{y})$
(c) $(\bar{x}, 0)$
(d) $(0, 0)$
›Reveal solutionSolution

A regression line always passes through the point of means $(\bar{x}, \bar{y})$ — option (a).

In the GSEB Class-12 Statistics Linear Regression chapter, the regression line of $Y$ on $X$ is written as:

$\hat{y} - \bar{y} = b_{yx}(x - \bar{x})$

…

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CBSE 2022Set MARCH1 markMCQ
$Q. The regression line always passes through which point?$
(a) $(\bar{x}, \bar{y})$
(b) $(0, \bar{y})$
(c) $(\bar{x}, 0)$
(d) $(0, 0)$
›Reveal solutionSolution

Each regression line is constructed to pass through the point of averages $(\bar{x}, \bar{y})$ , which is also where the two regression lines intersect.

…

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CBSE 2022Set MARCH1 mark
$Q. Give the name of a method to obtain the best fitted regression line.$
›Reveal solutionSolution

The best-fitted regression line is obtained by the Method of Least Squares.

Explanation. The method of least squares chooses the line for which the sum of the squares of the deviations (errors) of the observed values from the estimated values is minimum. This criterion yields …

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CBSE 2020Set MARCH1 markMCQ
$e$
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(b) $\hat{x}-\hat{y}$
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›Reveal solutionSolution

Error of estimation $e = y - \hat{y}$ (observed minus estimated) — option (a).

For the regression line of $Y$ on $X$ , we estimate $Y$ as $\hat{y} = a + bx$ . For a given observation the error (residual) of estimation is defined as the actual observed value of $Y$ minus the value estimated by the line:

$e = y - \hat{y}$

…

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CBSE 2020Set MARCH1 mark
$a$
›Reveal solutionSolution
$a$ = intercept ( $\hat{y}$ at $x = 0$ ); $b$ = regression coefficient $b_{yx}$ = slope = change in $\hat{y}$ per unit change in $x$ .

In the regression line of $Y$ on $X$ ,

$\hat{y} = a + bx,$

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$b$ is the regression coefficient of $Y$ on $X$ , denoted $b_{yx}$ : it is the slope of the line, i.e. the amount by which the estimated $\hat{y}$ changes for a one-unit increase in $x$ . …
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