Average rain and yield of a crop per acre in the y – Class 12 Statistics

Question 15 of 42

Average rain and yield of a crop per acre in the year of an orid region is given:

Particulars	Rainfall (cm)	Yield of crop (kg)
Mean	18	970
Standard Deviation	2	38

Correlation coefficient $= 0.6$

Obtain Regression line of yield on rain and estimate the yield of the crop if it rains 20 cms.

If $n = 10$ , $\Sigma x = 130$ , $\Sigma y = 220$ , $\Sigma x^2 = 2288$ , $\Sigma xy = 3467$ obtain regression line of $Y$ on $X$ and estimate value of $Y$ when $X = 16$ .

Gujarat GsebGujarat Board (GSEB) HSC Commerce Board 2020Subjective· 3mImportance★★★★★

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$b_{yx} = r\frac{S_y}{S_x} = 11.4$ ; $\hat{Y} = 764.8 + 11.4X$ ; at $X=20$ , yield $= 992.8$ kg. OR: $b_{yx} = 6070/5980 = 1.0151$ , $\hat Y = 8.80 + 1.0151X$ , at $X=16$ , $\hat Y = 25.05$ .

Main part — regression of yield ( $Y$ ) on rain ( $X$ ).

Given $\bar{X} = 18$ , $\bar{Y} = 970$ , $S_X = 2$ , $S_Y = 38$ , $r = 0.6$ .

Regression coefficient of $Y$ on $X$ :

$b_{yx} = r\cdot\frac{S_Y}{S_X} = 0.6\times\frac{38}{2} = 0.6\times 19 = 11.4$

Regression line of $Y$ on $X$ through the means:

$\hat{Y} - \bar{Y} = b_{yx}(X - \bar{X}) \implies \hat{Y} - 970 = 11.4(X - 18)$

$\hat{Y} = 970 + 11.4X - 205.2 = 764.8 + 11.4X$

Estimate yield when it rains $X = 20$ cm:

$\hat{Y} = 764.8 + 11.4(20) = 764.8 + 228 = 992.8 \text{ kg}$

OR — using raw sums ( $n=10$ , $\sum x = 130$ , $\sum y = 220$ , $\sum x^2 = 2288$ , $\sum xy = 3467$ ).

$\bar{x} = 13, \qquad \bar{y} = 22$ …

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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
$e$
(a) $y-\hat{y}$
(b) $\hat{x}-\hat{y}$
(c) $x-\hat{x}$
(d) $\hat{y}-y$
›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 $X$ ,

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

$a$ is the intercept constant: the value the line predicts for $\hat{y}$ when $x = 0$ . It fixes the position of the line on the $Y$ -axis.

$b$ 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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