To study the relationship between the time of usage of cars and its average annual maintenance cost of a car manufacturing company, the following information is obtained : | Car | 1 | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---|---| | Time of usage of a car (Years) $x$ | 3 | 1 | 2 | 2 | 5 | 3 | | Average annual maintenance cost (thousand ₹) $y$ | 10 | 5 | 8 | 7 | 13 | 8 | Obtain the regression line of $Y$ on $X$. Find an estimate of average annual maintenance cost when the usage of a car is 5 years. Also find its error.

> > [!TLDR] > $b_{yx} = \tfrac{108}{56} = 1.9286$, $a = 3.3571$; $\hat{y} = 3.3571 + 1.9286x$; at $x = 5$, $\hat{y} = 13.0$; error $= 13 - 13 = 0$. > > GSEB Class-12 Statistics, Linear Regression (line of $Y$ on $X$): > > Let $x$ = years of usage, $y$ = maintenance cost, $n = 6$. > > | $x$ | $y$ | $x^2$ | $xy$ | > |---|---|---|---| > | 3 | 10 | 9 | 30 | > | 1 | 5 | 1 | 5 | > | 2 | 8 | 4 | 16 | > | 2 | 7 | 4 | 14 | > | 5 | 13 | 25 | 65 | > | 3 | 8 | 9 | 24 | > | $\Sigma x = 16$ | $\Sigma y = 51$ | $\Sigma x^2 = 52$ | $\Sigma xy = 154$ | > > $\bar{x} = \dfrac{16}{6} = 2.6667$, $\bar{y} = \dfrac{51}{6} = 8.5$. > > **Regression coefficient of $Y$ on $X$:** > $$b_{yx} = \frac{n\Sigma xy - \Sigma x \, \Sigma y}{n\Sigma x^2 - (\Sigma x)^2} = \frac{6(154) - 16(51)}{6(52) - (16)^2} = \frac{924 - 816}{312 - 256} = \frac{108}{56} = 1.9286$$ > > **Intercept:** > $$a = \bar{y} - b_{yx}\,\bar{x} = 8.5 - 1.9286 \times 2.6667 = 8.5 - 5.1429 = 3.3571$$ > > **Regression line of $Y$ on $X$:** …

To study the relationship between the time of usage of cars and its average annual maintenance cost of a car m

Question 35 of 42

To study the relationship between the time of usage of cars and its average annual maintenance cost of a car manufacturing company, the following information is obtained :

Car	1	2	3	4	5	6
Time of usage of a car (Years) $x$	3	1	2	2	5	3
Average annual maintenance cost (thousand ₹) $y$	10	5	8	7	13	8

Obtain the regression line of $Y$ on $X$ . Find an estimate of average annual maintenance cost when the usage of a car is 5 years. Also find its error.

Gujarat GsebGujarat Board (GSEB) HSC Commerce Board 2025Subjective· 5mImportance★★★★★

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$b_{yx} = \tfrac{108}{56} = 1.9286$ , $a = 3.3571$ ; $\hat{y} = 3.3571 + 1.9286x$ ; at $x = 5$ , $\hat{y} = 13.0$ ; error $= 13 - 13 = 0$ .

GSEB Class-12 Statistics, Linear Regression (line of $Y$ on $X$ ):

Let $x$ = years of usage, $y$ = maintenance cost, $n = 6$ .

$x$	$y$	$x^2$	$xy$
3	10	9	30
1	5	1	5
2	8	4	16
2	7	4	14
5	13	25	65
3	8	9	24
$\Sigma x = 16$	$\Sigma y = 51$	$\Sigma x^2 = 52$	$\Sigma xy = 154$

$\bar{x} = \dfrac{16}{6} = 2.6667$ , $\bar{y} = \dfrac{51}{6} = 8.5$ .

Regression coefficient of $Y$ on $X$ :

$b_{yx} = \frac{n\Sigma xy - \Sigma x \, \Sigma y}{n\Sigma x^2 - (\Sigma x)^2} = \frac{6(154) - 16(51)}{6(52) - (16)^2} = \frac{924 - 816}{312 - 256} = \frac{108}{56} = 1.9286$

Intercept:

$a = \bar{y} - b_{yx}\,\bar{x} = 8.5 - 1.9286 \times 2.6667 = 8.5 - 5.1429 = 3.3571$

Regression line of $Y$ on $X$ : …

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

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CBSE 2026Set MARCH1 markMCQ
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The regression line passes through $(\bar x,\bar y)$ ; putting $\bar x=10$ gives $\bar y=15$ .

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

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CBSE 2022Set MARCH1 markMCQ
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