Tag: time series

Quizzes (7) Questions (24)

Questions Related to time series

Multiple choice statistics time series moving average and variation simple moving average

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

  1. $1$ only
  2. $1$ or $-1$
  3. $-1$ only
  4. $0$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $-1$
Multiple choice statistics time series moving average and variation simple moving average

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

$x$ $5$ $2$ $1$ $4$ $3$
$y$ $5$ $8$ $4$ $2$ $10$
  1. $y=-0.8x+7$
  2. $y=0.4x+7$
  3. $y=-0.4x+7$
  4. $y=-0.4x-7$
Reveal answer Fill a bubble to check yourself
C Correct answer
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, $y=-0.4x+7$
Multiple choice statistics time series moving average and variation simple moving average

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.

  1. $1$ only
  2. $2$ only
  3. Both $1$ and $2$
  4. Neither $1$ nor $2$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $(b)$.

Multiple choice statistics time series moving average and variation simple moving average

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?

  1. $y = 0.91 x + 8.74$
  2. $y = 1.02x + 8.74$
  3. $y = 1.02 x - 7.02$
  4. $y = 0.91 x - 7.02$
Reveal answer Fill a bubble to check yourself
B Correct answer
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.
Multiple choice statistics time series moving average and variation simple moving average

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 $y=5$

  1. $5.231$
  2. $4.591$
  3. $6.564$
  4. $8.231$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using the linear regression formula for x on y: x - mean_x = b_xy * (y - mean_y). Mean_x = 125/25 = 5. Mean_y = 100/25 = 4. Calculating b_xy and solving for x when y=5 yields approximately 4.591.

Multiple choice statistics time series moving average and variation simple moving average

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?

  1. 1 only

  2. 2 only

  3. Both 1 and 2

  4. Neither 1 nor 2
Reveal answer Fill a bubble to check yourself
D Correct answer
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.
Multiple choice statistics time series moving average and variation simple moving average

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 $14$ $20$
  1. $145.32$
  2. $44.23$
  3. $159.43$
  4. $561.12$
Reveal answer Fill a bubble to check yourself
C Correct answer
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
$y=1.143\times 70+79.43=159.43$

Multiple choice statistics time series moving average and variation simple moving average

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

$x$ $2$ $4$ $6$ $8$ $10$
$y$ $6$ $5$ $4$ $3$ $2$
  1. $x+ 2y=14$
  2. $x-2y=14$
  3. $2x+ y=14$
  4. $2x- 2y=14$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $x+2y=14$

Multiple choice statistics time series moving average and variation simple moving average

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.

  1. $12$ and $18$
  2. $18$ and $12$
  3. $13$ and $17$
  4. $17$ and $13$
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $17$.
Multiple choice statistics time series moving average and variation simple moving average

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 $x$.

  1. $4x-y-11=0$
  2. $7x-3y-18=0$
  3. $4x-y-2=0$
  4. $7x-3y-4=0$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $x$.