Question 12 Marks
The following measures are obtained to study the relation between rainfall in cm $(X)$ and yield of Bajri in Quintal per Hectare $(Y)$ in ten different regions during monsoon.
$n=10, \bar{x}=40, \bar{y}=175, s_{x}=12, \operatorname{Cov}(x, y)=360$
Obtain the regression line of yield $Y$ on rainfall $X.$
AnswerHere, $\operatorname{Cov}(x, y)=360$ and $s_x=12$
$\therefore s_x^2=144$
$\mathrm{b} =\frac{\operatorname{Cov}(x, y)}{s_x^2}$
$ =\frac{360}{144}$
$\therefore b =2.5$
$\text { and } a =\bar{y}-b \bar{x}$
$ =175-2.5(40)$
$ =175-100$
$\therefore a =75$
So, the regression line of $Y$ on $X$ is
$\hat{y} =a+b x$
$\therefore \hat{y} =75+2.5 x$
View full question & answer→Question 22 Marks
If $u=5(x-40), v=2(y-18)$ and $b_{y x}=1.6$, find the value of $b_{v u} .$
AnswerIf $u=x-A$ and $v=y-B$ then $b_{y x}=b_{v u}$
If $u=\frac{x-A}{c_{x}}$ and $v=\frac{y-B}{c_{y}}$ then $b_{y x}=b_{x} \cdot \frac{c_{y}}{c_{x}}$
Since $u=5(x-40)=\frac{x-40}{\frac{1}{5}}=\frac{x-A}{c_{x}}$ and $v=2(y-18)=\frac{y-18}{\frac{1}{2}}=\frac{y-B}{c_{y}}$
$ b_{y x}=b_{v u} \cdot \frac{c_{y}}{c_{x}}$
$\therefore b_{v u}=b_{y x} \cdot \frac{c_{x}}{c_{y}}=1.6 \times \frac{\left(\frac{1}{5}\right)}{\left(\frac{1}{2}\right)}=1.6 \times \frac{2}{5}=0.64 . $
View full question & answer→Question 32 Marks
If $u=10(x-4.5), v=\frac{y-50}{10}$ and $b_{y x}=0.25$, find the value of $b_{v u}$.
AnswerIf $u=x-A$ and $v=y-B$ then $b_{y x}=b_{v u}$
If $u=\frac{x-A}{c_{x}}$ and $v=\frac{y-B}{c_{y}}$ then $b_{y x}=b_{x} \cdot \frac{c_{y}}{c_{x}}$
Since $u=10(x-4.5)=\frac{x-4.5}{\frac{1}{10}}=\frac{x-A}{c_{x}}$ and $v=\frac{y-50}{10}=\frac{y-B}{c_{y}}$$b_{y r}=b_{w} \cdot \frac{c_{y}}{c_{x}} \quad \therefore b_{w}=b_{y x} \cdot \frac{c_{x}}{c_{y}}=0.25 \times \frac{\left(\frac{1}{10}\right)}{10}=0.25 \times \frac{1}{100}=0.0025$
View full question & answer→Question 42 Marks
If $u=\frac{x-5}{3}, v=\frac{y-8}{5}$ and $b_{y x}=0.9$, find the value of $b_{v u}$.
AnswerIf $u=x-A$ and $v=y-B$ then $b_{y x}=b_{v u}$
If $u=\frac{x-A}{c_{x}}$ and $v=\frac{y-B}{c_{y}}$ then $b_{y x}=b_{x} \cdot \frac{c_{y}}{c_{x}}$
Since $u=\frac{x-5}{3}=\frac{x-A}{c_{x}}$ and $v=\frac{y-8}{5}=\frac{y-B}{c_{y}}$$b_{y x}=b_{n u} \cdot \frac{c_{y}}{c_{x}} \quad \therefore b_{n u}=b_{y x} \cdot \frac{c_{x}}{c_{y}}=0.9 \times \frac{3}{5}=0.54$
View full question & answer→Question 52 Marks
If $b_{y x}=0.85, u=x-15$ and $v=y-20$, find the value of $b_{v u}$.
AnswerIf $u=x-A$ and $v=y-B$ then $b_{y x}=b_{v u}$
If $u=\frac{x-A}{c_{x}}$ and $v=\frac{y-B}{c_{y}}$ then $b_{y x}=b_{x} \cdot \frac{c_{y}}{c_{x}}$
Since $u=x-15=x-A$ and $v=y-20=y-B$$\therefore b_{v u}=b_{y x}=0.85$
View full question & answer→Question 62 Marks
If one observation $(10,30)$ is used in the fitting of the line $\hat{y}=22+0.8 x$, find the error in estimating $Y$ for $X=10$. What can you deduce from the value of the error ?
View full question & answer→Question 72 Marks
The fitted regression line of $Y$ on $X$ is $\hat{y}=50+3.5 x$. If an observation $(16,108)$ is used in fitting of the line, find the error in estimating $Y$ for $X=16$.
View full question & answer→Question 82 Marks
If the regression line of $Y$ on $X$ is $\hat{y}=11.5+0.65 x$ and $\bar{y}=18$, find $\bar{x}$.
View full question & answer→Question 92 Marks
If the regression line of $Y$ on $X$ is $\hat{y}=12-1.5 x$ and the mean of $X$ is 6 , find the mean of $Y$.
View full question & answer→Question 102 Marks
If $\bar{x}=10, \bar{y}=25, \Sigma(x-10)(y-25)=120$ and $\Sigma(x-10)^2=100$, find the values of $a$ and $b$ for the regression line of $Y$ on $X$.
Answer$\bar{x}=10 ; \bar{y}=25 ;$
$ \Sigma(x-10)(y-25)=120, \Sigma(x-10)^2=100$
$ \bar{x}=10 \text { and } \bar{y}=25 .$
$ \therefore \Sigma(x-10)(y-25)=\Sigma(x-\bar{x})(y-\bar{y})=120 \text { and } \Sigma(x-10)^2=\Sigma(x-\bar{x})^2=10$
$ \text { Now, } b=\frac{\Sigma(x-\bar{x})(y-\bar{y})}{\Sigma(x-\bar{x})^2}=\frac{120}{100}=1.2$
$ a=\bar{y}-b \bar{x}$
$ \text { Putting } \bar{x}=10, \bar{y}=25 \text { and } b=1.2 \text {, we get }$
$ a=25-1.2(10)=25-12=13$
Hence, $a$ and $b$ obtained for the regression line of $Y$ on $X$ obtained are $13$ and $1.2$ respectively.
View full question & answer→Question 112 Marks
For the regression line given in the previous question $(7)m$ if the value of $Y$ is to be increased by $10$ units, how many units should be increased in the value of $X ?$
Answer$\hat{y}=35+2 \bar{x}$ is given.
Here, $b=2$
Hence, if the value of $Y$ is to be increased by $10$ units, then $\frac{10}{2}=5$ units should be increased in the value of $X$.
View full question & answer→Question 122 Marks
If the regression line of $Y$ on $X$ is $ŷ = 35 + 2x$ and Cov $(x, y) = 50,$ find the standard deviation of $X.$
Answer$\hat{y}=35+2 x$
$\therefore b=2, \operatorname{Cov}(x, y)=50,$
$ S_x=?$
Now, $\mathrm{b}=\frac{\operatorname{Cov}(x, y)}{\mathrm{S}_x^2}$
$\therefore 2=\frac{50}{\mathrm{~s}_x^2}$
$ \therefore 2 \mathrm{~S}_{\mathrm{x}}{ }^2=50$
$ \therefore \mathrm{S}_{\mathrm{x}}{ }^2=\frac{50}{2}=25$
$ \therefore \mathrm{S}_{\mathrm{x}}=5$
Hence, Standard deviation of $X$ obtained is $5 .$
View full question & answer→Question 132 Marks
If the regression coefficient of the regression line of $Y$ on $X$ in $0.6$ and the standard deviations of $X$ and $Y$ are $5$ and $3$ respectively, find the coefficient of determination.
Answer$b=0.6 ; S_x=5, S_y=3, r=?$
Now, $b=r \cdot \frac{s_y}{\mathrm{~S}_x}$
$\therefore 0.6=r \cdot \frac{3}{5}$
$ \therefore 0.6=0.6 r$
$ \therefore r=\frac{0.6}{0.6}=1$
Coefficient of determination $R=(r)^2=(1)^2=1$
View full question & answer→Question 142 Marks
If $b = 1.5, r = 0.8 $ and standard deviation of $X$ is $1.6,$ find the standard deviation of $Y.$
Answer$
b=1.5, r=0.8, S_x=1.6, S_y=?
$
Now, $b=r \cdot \frac{\mathrm{S}_y}{\mathrm{~S}_x}$
$
\begin{aligned}
& \therefore 1.5=0.8 \times \frac{S_y}{1.6} \\
& \therefore 1.5=\frac{s_y}{2} \\
& \therefore 1.5=1.5 \times 2=3
\end{aligned}
$
View full question & answer→Question 152 Marks
Interpret $\mathrm{b}_{\mathrm{yx}}=5$.
AnswerIf $b_{x y}=5$ then it can be told that due to increase of $1$ unit in the value of variable $X$, value of variable $Y$ will be increased $5$ unit.
View full question & answer→Question 162 Marks
If $x̄ = 30, ȳ = 20$ and $b = 0.6,$ find the intercept of the regression line of $Y$ on $X$ and write equation of the line.
Answer$X̄ = 30, ȳ = 20, b = 0.6$
Intercept $a = ȳ – bx̄$
$\therefore a = 20 – 0.6(30) = 20 – 18 = 2$
Putting $a = 2$ and $b = 0.6$ in $ŷ = a + bx,$ the equation of the regression line obtained is $ŷ = 2 + 0.6x.$
View full question & answer→Question 172 Marks
If $b_{y x}=0.75, u=6(x-20)$ and $v=2(y-15)$ for the data in the study of a regression line then find the value of $b_{v u}$.
Answer$b_{y x}=0.75 ;$
$u=6(x-20)=\frac{(x-20)}{1 / 6} \quad \therefore \mathrm{C}_x=\frac{1}{6}$
$v=2(y-15)=\frac{y-15}{1 / 2} \quad \therefore \mathrm{C}_y=\frac{1}{2}$
Now, $b_{u v}$$=b_{y x} \div \frac{\mathrm{C}_y}{\mathrm{C}_x} \quad$
$=b_{y x} \cdot \frac{\mathrm{C}_x}{\mathrm{C}_y}$
$=0.75 \times \frac{\frac{1}{6}}{\frac{1}{2}}$
$=0.75 \times \frac{1}{6} \times \frac{2}{1}$
$=\frac{0.75}{3}=0.25$
View full question & answer→Question 182 Marks
What are the constants a and b in the regression line $\hat{y} = a + bx ?$
AnswerIn the regression line $\hat{y} = a + bx $ a is known as intercept of regression line and $b$ is known as regression co-efficient or slope of regression line.
View full question & answer→Question 192 Marks
If $\bar{x}=30, \bar{y}=35, \sum(x-30)(y-35)-225$ and $\sum[x-30]^{2}=90$, find the values of $a$ and $b$ for the regression line of $y$ on $X$.
View full question & answer→Question 202 Marks
If $\hat{y}=-35+4 x$ and standard deviation of $x$ is $6 ,$ find Coy $(x, y)$.
View full question & answer→Question 212 Marks
If the regression line of $Y$ on $X$ is $\hat{y}=120+0.4 X$ and $\operatorname{Cov}(X, y)=48.4$, find the standard deviation of $X$.
Answer$\overline{ S }=11$
View full question & answer→Question 222 Marks
If coefficient of determination is $0.81$ and $S_{x}: S_{y}=3: 5$, find the value of regression on ceefficient.
View full question & answer→Question 232 Marks
If the slope of the regression line of $Y$ on $X$ is $0.5$ and the variance of $X$ and $Y$ are $576$ and $225$ respectively, find the coefficient of determination.
Answer$r=0.8, R^{2}=0.64$
View full question & answer→Question 242 Marks
If the regression line of $Y$ on $X$ is $\hat{y}=25-1.8 x$ and the mean of $X$ is $5 ,$ find the meanof $Y$.
View full question & answer→Question 252 Marks
If $b_{y z}=0.90,4=5(x-30)$ and $v=3(y-20)$ for the data in a study of a regression linethen find the value of $b_{v c}$.
View full question & answer→Question 262 Marks
If $\sum(x-40)=0, \sum(y-50)=0, \sum(x-40)(y-50)=200, \sum\left[x-40^{2}\right.$ $=250$ and $\sum[x-50]^{2}-400$, find the values of a and $b$ for the regression line of $Y$ on $x$.
Answer$\bar{b}=0.8, a =18$
View full question & answer→Question 272 Marks
Interpret: $b=-5$
Answer$b = -5.$ So, it can be said that because of increase of $1$ unit in $x,$ there is appropriate $5$ units of decrease in $y.$
View full question & answer→Question 282 Marks
Interpret: $b_{y z}=6$
Answer$b_{y z}=6$
So, it can be said that because of increase of $1$ unit in $x,$ there is appropriate $6$ units of increase in $y.$
View full question & answer→Question 292 Marks
The fitted regression line of $Y$ on $X$ is $\hat{y}=80+4.5 X$ and one of the observations used in fitting of the line is $(20,170)$. Find the error in estimating $Y$ for $X=20$ and interpret it.
AnswerError $e = 0;$ Points $(20, 170)$ is on the fitted line.
View full question & answer→Question 302 Marks
The fitted regression line of $Y$ on $X$ is $\hat{y}=80+2.5 X$ and one of the observations used in fitting of the line is $(12,112)$. Find the error in estimating $Y$ for $X=12$.
View full question & answer→Question 312 Marks
$B= 80 + 2.5x$ and one of the observations used in fitting of the line is $(12, 112).$ Find the error in estimating $Y$ for $X = 12.$
View full question & answer→Question 322 Marks
Differentiate the dependent and independent variables from the followings : $(1)$ Rain and production $(5)$ Consumption of electricity and its hill. $(2)$ Prom and investment $(6)$ Marks in exams and reading time. $(3)$ Yield of crop and use of fertilizer. $(7)$ Income and saving $(4)$ Advertisement expense and sales. $(8)$ Number of vehicles and number of accidents.
AnswerIndependent Variable :
- $(1)$ Rain $(2)$ Investment $(3)$ Use of fertilizer $(4)$ Advertisement expense $(5)$ Consumption of electricity $(6)$ Reading time $(7)$ Income $(8)$ Number of vehicles.
Dependent Variable :
- $(1)$ Production $(2)$ Profit $(3)$ Yield of crop $(4)$ Sales $(5)$ Bill of electricity $(6)$ Marks in exams $(7)$ Saving $(8)$ Number of accidents
View full question & answer→Question 332 Marks
State the uses of two regression lines.
AnswerFollowing are the uses of two regression lines.
$(1)$ To estimate the value of a dependent variable for a given value of independent variable. i.e. as prediction formula.
$(2)$ To estimate the probable change in the value of a dependent variable for a unit change in the value of independent variable.
View full question & answer→Question 342 Marks
If $\bar{x}=30, \bar{y}=20$ and $b=0.5$, find the intercept of the regression line of Y on X and write equation of the line.
AnswerIntercept $a = \bar{y} - b\bar{x} = 20 - (0.5)(30) = 20 - 15 = 5$.
Equation : $\hat{y} = a + bx \implies \hat{y} = 5 + 0.5x$.
View full question & answer→Question 352 Marks
If $b=1.5$, $r=0.8$ and standard deviation of X is 1.6, find the standard deviation of Y.
Answer$b_{yx} = 1.5$, $r = 0.8$, $S_{x} = 1.6$.
We know that $b_{yx} = r \cdot \frac{S_{y}}{S_{x}}$.
$1.5 = 0.8 \cdot \frac{S_{y}}{1.6}$
$1.5 = 0.5 \cdot S_{y}$
$S_{y} = \frac{1.5}{0.5} = 3$.
Therefore, the standard deviation of Y is 3.
View full question & answer→Question 362 Marks
If $ \overline{x}=5, \overline{y}=11 $ and $ b=1.2 $, obtain the regression line Y on X.
AnswerCalculate intercept $ a $ :
$ a = \overline{y} - b\overline{x} = 11 - 1.2(5) = 11 - 6 = 5 $
Regression line : $ \hat{y} = a + bx $
$ \hat{y} = 5 + 1.2x $
View full question & answer→Question 372 Marks
If $b_{yx}=0.75$, $u=6(x-20)$ and $V=2(y-15)$ for the data in the study of a regression line then find the value of $b_{Vu}$.
AnswerHere, $u = \frac{x-20}{1/6} \implies C_x = 1/6$.
$V = \frac{y-15}{1/2} \implies C_y = 1/2$.
$b_{yx} = b_{Vu} \times \frac{C_y}{C_x}$
$0.75 = b_{Vu} \times \frac{1/2}{1/6}$
$0.75 = b_{Vu} \times 3 \implies b_{Vu} = 0.75 / 3 = 0.25$.
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