From the following information, obtain the regression line of monthly expenditure on monthly income. If Namrat

Question 16 of 42

From the following information, obtain the regression line of monthly expenditure on monthly income. If Namrata's monthly income is ₹75 thousand estimate her monthly expenditure:

Monthly income (thousand)	60	70	64	68	62	65	72
Monthly Expenditure (thousand)	50	59	57	50	53	58	60

Gujarat GsebGujarat Board (GSEB) HSC Commerce Board 2020Subjective· 5mImportance★★★★★

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$b_{yx} = 471/790 = 0.596$ ; $\hat y = 16.02 + 0.596x$ ; at income 75, expenditure $\approx$ ₹60.74 thousand.

Let $x$ = monthly income and $y$ = monthly expenditure (both in ₹ thousand). Use assumed means $A = 65$ , $B = 55$ , with $d_x = x-65$ , $d_y = y-55$ :

$x$	$y$	$d_x$	$d_y$	$d_xd_y$	$d_x^2$
60	50	-5	-5	25	25
70	59	5	4	20	25
64	57	-1	2	-2	1
68	50	3	-5	-15	9
62	53	-3	-2	6	9
65	58	0	3	0	0
72	60	7	5	35	49
Σ		6	2	69	118

$n = 7$ , so $\bar{x} = 65 + \frac{6}{7} = 65.857$ , $\bar{y} = 55 + \frac{2}{7} = 55.286$ .

Regression coefficient of $y$ on $x$ :

$b_{yx} = \frac{n\sum d_xd_y - \sum d_x\sum d_y}{n\sum d_x^2 - (\sum d_x)^2} = \frac{7(69) - (6)(2)}{7(118) - (6)^2} = \frac{483 - 12}{826 - 36} = \frac{471}{790} = 0.5962$

Regression line of expenditure on income: …

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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
$Q. The regression line always passes through which point?$
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›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
$Q. The regression line always passes through which point?$
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(b) $(0, \bar{y})$
(c) $(\bar{x}, 0)$
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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
$Q. Give the name of a method to obtain the best fitted regression line.$
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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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Error of estimation $e = y - \hat{y}$ (observed minus estimated) — option (a).

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…

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

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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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