Tag: time series

Questions Related to time series

The regression lines will be perpendicular to each other if the coefficient of correlation r is equal to.

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  1. $1$ only

  2. $1$ or $-1$

  3. $-1$ only

  4. $0$

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Correct Option: B
Explanation:
The two regression lines are perpendicular to each other.

When coefficient of correlation is perfectly positive or negative $r$ = The two regression lines coincide 
Therefore the answer will be $1$ or $-1$

Find the equation of $y$ on $x$ on the basis of the following data:

$x$ $5$ $2$ $1$ $4$ $3$
$y$ $5$ $8$ $4$ $2$ $10$
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  1. $y=-0.8x+7$

  2. $y=0.4x+7$

  3. $y=-0.4x+7$

  4. $y=-0.4x-7$

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Correct Option: C
Explanation:
 $x$  $y$  $xy$  $x^2$
 5  5  25  25
 2  8  16  4
 1  4  4  1
 4  2  8  16
 3  10  30  9
 $\sum x=15$  $\sum y=29$  $\sum xy=83$  $\sum x^2=55$

The linear equation be y=a+bx

$n$ is the number of observations
$n=5$
where $a=\dfrac{(\sum y)(\sum x^2)-(\sum x)(\sum xy)}{n(\sum x^2)-(\sum x)^2}$
$\implies a=\dfrac{(29)(55)-(15)(83)}{5(55)-(15)^2}=\dfrac{350}{50}=7$
and $b=\dfrac{n(\sum xy)-(\sum x)(\sum y)}{n(\sum x^2)-(\sum x)^2}$
$\implies b=\dfrac{(5)(83)-(15)(29)}{5(55)-(15)^2}=\dfrac{-20}{50}=-0.4$
Therefore, $y=-0.4x+7$

Which of the following statements is/are correct in respect of regression coefficients?
$1.$ It measures the degree of linear relationship between two variables.
$2.$ It gives the value by which one variable changes for a unit change in the other variable.
Select the correct answer using the code given below.

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  1. $1$ only

  2. $2$ only

  3. Both $1$ and $2$

  4. Neither $1$ nor $2$

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Correct Option: A
Explanation:

When the regression line is linear $(y = ax + b)$ the regression coefficient is the constant $(a)$ that represents the rate of change of one variable $(y)$ as a function of changes in the other $(x)$ i.e. it is the slope of the regression line. Hence, it measures relationship between 2 variables but does not always account for exact change in value of 1 variable due to unit change in another variable due to non-zero value of $(b)$.

For 10 observations on price (x) and supply (y), the following data was obtained: $\sum x = 130, \sum y = 220, \sum x^2 = 2288, \sum y^2 = 5506$ and $\sum xy = 3467$. 
What is the line of regression of y on x?

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  1. $y = 0.91 x + 8.74$

  2. $y = 1.02x + 8.74$

  3. $y = 1.02 x - 7.02$

  4. $y = 0.91 x - 7.02$

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Correct Option: B
Explanation:
Line of regression of $y$ on $x$ is : $y - \overline{y} = b_{yx} (x - \overline{x})$ where $\overline{y}$ and $\overline{x}$ are mean values.
$\therefore \overline{y}=22, \overline{x}=13$ as $n=10$

Also, $b_{yx}=r\dfrac{\sigma_y}{\sigma_x}$

$\therefore r= \dfrac{n\sum-(\sum x)(\sum y)}{\sqrt{[n\sum x^2-(\sum x)^2][n\sum y^2-(\sum y)^2]}}=0.962$

$\sigma_y=8.2$ and $\sigma_x=7.73$

$\therefore b_{yx}=1.02$

$\therefore y=1.02x+8.74$ is the required line of regression.

For two variables $x$ and $y$ ,the following data are given as
$\Sigma x=125,\Sigma y=100, \Sigma x^2=1650,\Sigma y^2=1500,\Sigma xy=50,n=25$.Find the value of $x$ when $y=5$

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  1. $5.231$

  2. $4.591$

  3. $6.564$

  4. $8.231$

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Correct Option: B

Consider the following statements: (1) If the correlation coefficient ${ r } _{ xy }=0$, then the two lines of regression are parallel to each other (2) If the correlation coefficient ${ r } _{ xy }=+1$, then the two lines of regression are perpendicular to each other? Which of the above statements is/are correct?

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  1. 1 only

  2. 2 only

  3. Both 1 and 2

  4. Neither 1 nor 2

Show answer
Correct Option: D
Explanation:
If $r = 0\implies$ lines do not have anything common and hence, lines of regression are perpendicular.
when $r = 1\implies$ then the lines superimpose one another and hence, lines of regression are parallel/co-incident. So, both statements are wrong.

Find the Value of $y$ from the following data when $x=70$ and coefficient of correlation $0.8$.

Series $x$ $y$
A.M $18$ $100$
Standard Deviation $14$ $20$
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  1. $145.32$

  2. $44.23$

  3. $159.43$

  4. $561.12$

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Correct Option: C
Explanation:

$\bar x=18$, $\bar y=100$, $\sigma_x=14$, $\sigma_y=20$ and $r=0.8$
Regression line of y on x will be $y-\bar y=r\dfrac{\sigma_y}{\sigma_x}(x-\bar x)$
Subsitute all the abpve value in an equation of regression line, we get
$y-100=\dfrac{0.8\times 20}{14}(x-18)$
$y-100=1.143(x-18)$
$y=1.143x+79.43$
Now, Subsitute the value of x=70, we get
$y=1.143\times 70+79.43=159.43$

Find the equation of $x$ on $y$ on the basis of the following data:

$x$ $2$ $4$ $6$ $8$ $10$
$y$ $6$ $5$ $4$ $3$ $2$
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  1. $x+ 2y=14$

  2. $x-2y=14$

  3. $2x+ y=14$

  4. $2x- 2y=14$

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Correct Option: A
Explanation:

Let the equation be $ax+by+c=0$


At $x=2 , y=6$
$\Rightarrow 2a+6b+c=0$

At $x=4, y=5$
$\Rightarrow 4a+5b+c=0$

At $x=6, y=4$
$\Rightarrow 6a+4b+c=0$

Solving these, we get
$a=1$
$b=$4
$c=-14$

$\therefore$ Equation is $x+2y=14$

If $4\bar {x}-5\bar y+33=0$ and $20\bar x-9\bar y=107$ are two lines of regression, then what are the values of $\bar { x } $ and $\bar { y } $ respectively.

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  1. $12$ and $18$

  2. $18$ and $12$

  3. $13$ and $17$

  4. $17$ and $13$

Show answer
Correct Option: C
Explanation:
The given equations are $4\overline{x} - 5\overline{y} + 33 = 0$ .... $(i)$ and $20\overline{x} - 9\overline{y} -107 = 0$ ..... $(ii)$
Solving $(i)$ and $(ii)$, we get
$\overline{x} = 13$ 
Substituting $x$ in $(i)$, we get
$\overline{y} =17$
Hence, the answer is $13$ and $17$.

For the variables $x$ and $y$, the regression equations are given as $7x-3y-18=0$ and $4x-y-11=0$. Identify the regression equation of $y$ on $x$.

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  1. $4x-y-11=0$

  2. $7x-3y-18=0$

  3. $4x-y-2=0$

  4. $7x-3y-4=0$

Show answer
Correct Option: B
Explanation:

Let us assume that $7x-3y-18=0$ is the regression equation of $y$ on $x$.


Consider $7x-3y-18=0$

$\Rightarrow y=-6+\dfrac{7}{3}x$ 

$\therefore b_{yx}=\dfrac{7}{3}$

Now consider $4x-y-11=0$

$\Rightarrow x=\dfrac{11}{4}+\dfrac{1}{4}y$

$\therefore b_{xy}=\dfrac{1}{4}$

Now taking the product, $b_{yx} \times b_{xy}=\dfrac{7}{3} \times \dfrac{1}{4}=\dfrac{7}{12}<1$

Since the product is less than one, our assumptions are correct.

Thus $7x-3y-18=0$ is the regression equation of $y$ on $x$.