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Statistics 3 Marks Each

PART 1 : LINEAR REGRASSION · Gujarat Board (GSEB) STD 12 Commerce (GSEB - English Medium). 9 questions.

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Question 13 Marks
The following results are obtained while studying the two related variables $X$ and $Y$. Obtain the regression line of $Y$ on $X$ from it. $n=10, \Sigma x y=17880, \Sigma x=300, \bar{y}=50$ and $S_x=12$ If $X=40$ find $\hat{y}$.
Answer
$\hat{y}=-10+2 x$, For $x=40, \hat{y}=70$
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Question 23 Marks
Answer the following questions from the given data.
$(i)$ If $b_{y x}=0.8, u=x-10, v=y+15$ then find the value of $b_{v i r}$.
$(ii)$ If $u=\frac{x-5}{3}, v=\frac{y-8}{5}$ and $b_{v u}=0.54$ then find the value of $b_{y x}$.
$(iii)$ If $u=5(x-40), v=\frac{y-50}{10}$ and $b_{y x}=0.25$ then find the value of $b_{ v }$.
Answer
$(i)\ b_{v u}=0.8$
$(ii)\ b_{y x}=0.9$
$(iii)\ b_{v u}=0.005$
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Question 33 Marks
The regression line of $Y$ on $X$ is $6 x+8 y-64=0$ and the variance of $Y$ is $4$ times of the variance of $X$. If the value of $X$ changes by $4$ units, what will be the effect on the value of $Y\ ?$ Also find $R^2$ and interpret it.
Answer
For a change of $4$ units in the value of $x$, the value of $y$ will decrease by $3$ units. $R^2=0.14$, The assumption of linear regression is not valid
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Question 43 Marks
If the regression line of $Y$ on $X$ is $\hat{y}=38+2 x, \operatorname{Cov}(x, y)=32$ and variance of $Y=100$, find coefficient of determination and interpret it.
Answer
$R^2=0.64$, The assumption of linear regression is valid
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Question 53 Marks
If the regression line of $Y$ on $X$ is $\hat{y}=50-0.6 x$ and one of the observations used in fitting of the line is $(15,41)$, find the error for the estimated value and what can be concluded from this error ?
Answer
$\hat{y}=41, e=0$, All the points are on the line
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Question 63 Marks
If the regression line of $Y$ on $X$ is $\hat{y}=a-0.2 x$ and $\bar{x}=100, \bar{y}=85, S_x^2=900$ and $S_y^2=100$ then $(i)$ Find the value of $a\ (ii)$ Find coefficient of determination and interpret it.
Answer
$(i) \ a=105$
$(ii)\ R^2=0.36$, The assumption of linear correlation is not valid
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Question 73 Marks
The following information is obtained while studying the relation between the carpet area $($square meter$)$ and the monthly rent $($thousand $₹)$ in a city. Obtain the regression line of the monthly rent on the carpet area.
$n=50, \Sigma x=650, \Sigma y=750, \Sigma x y=10000, \Sigma x^2=8800$ and $\Sigma y^2=11000$
Answer
$\hat{y}=5.77+0.71 x$
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Question 83 Marks
The following results are obtained while inquiring the relation between the time of study $($in hours$)$ and the percentage obtained by $8$ students of Standard $12.$ Obtain the regression line of $Y$ on $X$ from it. $\Sigma x=1346, \Sigma y=1338, \Sigma(x-168)=2, \Sigma(y-167)=2, \Sigma(x-168)^2=50$ and $\Sigma(x-168)(y-167)=34$
Answer
$\hat{y}=52.84+0.68 x$
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Question 93 Marks
The following data is obtained while inquiring $10$ stationery stores for the price of fullscape book $(₹)$ and their sales $($units$).$ Obtain the regression line of sales on the price of fullscape book from it.
$\Sigma x=145, \Sigma y=260, \Sigma(x-14.5)^2=650, \Sigma(y-26)^2=440$ and $\Sigma(x-14.5)(y-26)=418$
Answer
$\hat{y}=16.72+0.64 x$
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3 Marks Each - Statistics STD 12 Commerce Questions - Vidyadip