### Moving average method - Class IX

 Description: moving average method Number of Questions: 37 Created by: Mira Shah Tags: maths time series business maths statistics time series and forecasting linear regression regression analysis analyzing data moving averages
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A train running at $\dfrac{7}{{11}}$ of its own speed reached a place in 22 hours. How much time could be saved if the train would have run at its own speed ?

1. 7 hours

2. 8 hours

3. 14 hours

4. 16 hours

Correct Option: B
Explanation:
Let s be the actual speed of the train and d be the distance that needs to be covered to reach that place.

Here, speed$=\dfrac{Distance}{Time}$

As per question,

The train is running at $\dfrac{7}{11}$ of its own speed and reaches the place in $22$ hrs.

$\therefore d=\dfrac{7}{11}s.22$

$\Rightarrow d=14s$ ………..$(1)$

Let t be the time taken by the train to reach the place with its actual speed.

$\therefore d=st$ ………..$(2)$

from $(1)$ and $(2)$, we get

$14s=st$

$\Rightarrow t=14$ hours

$\therefore$ Time that could be saved$=22-14$

$=8$hrs.

The total numbers of squares on a chessboard is

1. $206$

2. $205$

3. $204$

4. $202$

Correct Option: C
Explanation:

From Permutation and Combination,

no. of squares of 1x1 $=1^2$
no. of squares of 2x2 $=2^2$
no. of squares of 3x3$=3^2$
.
.
.
no. of squares of 8x8 $=8^2$

Hence Total number of squares $=1^2+2^2+......+8^2$
$\implies 204$

The average weight of $A,B,C$ is $45kg$ . If the average weight of $A$ and $B$ be $40kg$ and that of $B$ and $C$ be $43kg$, find the weight of $B$ ?

1. $21kg$

2. $20kg$

3. $38kg$

4. $31kg$

Correct Option: D
Explanation:

$\cfrac{A+B+C}{3}=45$ $kg$

$\Rightarrow A+B+C=45\times3=135$ $kg$
$\Rightarrow \cfrac{A+B}{2}=40$ $kg$
$\Rightarrow A+B=80$ $kg$
$\therefore C=135-80=55$ $kg$
$\Rightarrow \cfrac{B+C}{2}=43$ $kg$
$\Rightarrow B+C=86$ $kg$
$\therefore B=86-55=31$ $kg$
$\therefore$ The weight of $B$ is $31$ $kg$.

In a call of 100 students there are 70 boys whose average marks in a subject is 75. If the average marks of the complete class is 72 . then the average marks of the girls is

1. 73

2. 65

3. 68

4. 74

Correct Option: B
Explanation:
Total student = 100
For 70 students total marks = 75×7075×70
=5250=5250
Total marks of girls = & 7200-5250$=1950=1950 Average of girls = 195030195030 =65 .Twelve years hence a man will be four times as he was twelve years ago, then his present age is 1. 20 years 2. 25 years 3. 28 years 4. 30 years Correct Option: A Explanation: Let his present age be x According to problem x + 12 = 4 (x - 12) -3x = - 48 - 12 3x = 60 x = 20 years Moving-averages: 1. Give the trend in a straight line 2. Measure the seasonal variations 3. Smooth-out the time series 4. None of them Correct Option: C Explanation:$\Rightarrow$Moving-averages :$Smooth-out\,the\,time\,series$.$\Rightarrow$A moving average is a widely used indicator in technical analysis that helps smooth out price action by filtering out the “noise” from random price fluctuations.$\Rightarrow$It is a trend-following, or lagging, indicator because it is based on past prices. The problem given below consists of a question followed by three statements. You have to study the question and the statements and decide which of the statements(s) is / are necessary to answer the question. How many marks did Tarun secure in English? (i)The average marks obtained by Tarun in four subjects including English are 60. (ii) The total marks obtained by him in English and Mathematics together are 170. (iii)The total marks obtained by him in Mathematics and science together are 180. 1. I and II only 2. II and III only 3. I and III only 4. None of these Correct Option: D Explanation: These statements are not enough to find out his English marks. So Ans = D Choose the correct answer from the alternatives given. A man covers a distance of$160$km at$64 km/hr$and next$160$km at$80 km/hr$. What is his average speed for his whole journey of$320$km? 1.$73.00km/h$2.$71.11km/h$3.$70.50km/h$4.$72.25km/h$Correct Option: B Bira and his wife Sheena have two daughters aged$12$and$16$. Sheena's mother and father, aged$65$and$72$, also live with them. Bira is currently looking for work, but can't find any. His elder daughter completed class$10$and prefers to look for work. Sheena prefers to stay at home to look after house works. How many unemployed members does Bira's family have? 1.$1$2.$2$3.$3$4.$4$Correct Option: B Explanation: Only Bira and his elder daughter can be called unemployed as per the given situation. A qualitative forecast 1. predicts the quality of a new product. 2. predicts the direction, but not the magnitude, of change in a variable. 3. is a forecast that is classified on a numerical scale from$1$(poor quality) to$10$(perfect quality). 4. is a forecast that is based on econometric methods. Correct Option: B Explanation: Qualitative forecast is an estimation methodology which uses expert judgment, rather than numerical analysis. It predicts the direction, but not the magnitude of change in variable. Two pipes A and B can fill a tank in 20 and 30 minutes respectively. If both the pipes are used together, then how long will it take to fill the tank? 1.$12 min$2.$15 min$3.$25 min$4.$50 min$Correct Option: A Explanation:$\displaystyle \frac{1}{20} + \frac{1}{30} = \frac{1}{x}\displaystyle \frac{5}{60} = \frac{1}{x}     \Rightarrow 12 min$The first step in time-series analysis is to 1. perform preliminary regression calculations. 2. calculate a moving average. 3. plot the data on a graph. 4. identify relevant correlated variables. Correct Option: C Explanation: The first step in time series analysis is to plot the data on a graph. In moving average method, we cannot find the trend values of some: 1. Middle periods 2. End periods 3. Starting periods 4. Between extreme periods Correct Option: D Explanation: In moving average method, we cannot find the trend value of some:between extreme periods. The method of least squares dictates that we choose a regression line where the sum of the square of deviations of the points from the line is: 1. Maximum 2. Minimum 3. Zero 4. Positive Correct Option: B Explanation:$\Rightarrow$The method of least squares dictates that we choose a regression line where the sum of the square of deviations of the points from the line is$:Minimum\Rightarrow$A process by which we estimate the value of dependent variable on the basis of one or more independent variables is regression.$\Rightarrow$More specifically, regression analysis helps one understand how the typical value of the dependent variable (or 'criterion variable') changes when any one of the independent variables is varied, while the other independent variables are held fixed. In simple linear regression, the numbers of unknown constants are: 1. One 2. Two 3. Three 4. Four Correct Option: B Explanation:$\Rightarrow$In simple linear regression, the number of unknown constants are$:Two\Rightarrow$The line of regression of$y$on$x$is given by$y=a+bx$where$a$and$b$are unknown constants known as intercept and slope of the equation. This is used to predict the unknown value of variable$y$when value of variable$x$is known. If one regression coefficient is greater than one, then other will be: 1. Less than one 2. More than one 3. Equal to one 4. None of these Correct Option: A Explanation: Both the regression coefficients$(b_{xy},b_{yx})$must have the same sign. i.e., if one of them is positive other should positive or if one of them is negative other should be negative. If one regression coefficient is greater than one, then other coefficient should be less than one. The purpose of simple linear regression analysis is to: 1. Predict one variable from another variable 2. Replace points on a scatter diagram by a straight-line 3. Measure the degree to which two variables are linearly associated 4. Obtain the expected value of the independent random variable for a given value of the dependent variable Correct Option: A Explanation: The regression model gives the relation between two or more variables. The linear regression model gives the relation between two or more variables using a straight line. Using the linear regression analysis we can estimate the value of one variable using another variable. Ayushi used the data from a scatterplot to determine a regression model showing the relationship between the population in the area where she lived and the number of years,$x$, after she was born. The result was an exponential growth equation of the form$y={x} _{0}{\left(1+r\right)}^{x}$. Then${x} _{0}$most likely represents 1. The population in the year that she was born 2. The rate of change of the population over time 3. The maximum population reached during her lifetime 4. The number of years after her birth when the population reached its maximum Correct Option: A The independent variable in a regression line is: 1. Non-random variable 2. Random variable 3. Qualitative variable 4. None of these Correct Option: A Explanation: A linear regression line has an equation of the form$Y=a+bX$where$Y$is called dependent variable or response.$X$is called independent variable or predictors or explanatory variable$a$is the y-intercept and$b$is the slope of the line. The independent variable is a non-random variable. The non-random variable don't admit a probability measure. In regression analysis, if observed cost value is$50$and predicted cost value is$7$then disturbance term is 1.$53$2.$37$3.$43$4. None of these Correct Option: C Explanation: Given that observed value is$50$and predicted value is$7$The disturbance term is$observed - predicted=50-7=43$State true or false: The coefficient of correlation between two variables$x$and$y$is:$r=\cfrac { { \sigma }^{ 2 }x+{ \sigma }^{ 2 }y-y }{ 2{ \sigma }_{ x }{ \sigma } _{ y } } $1. True 2. False Correct Option: A Explanation: Coefficient of correlation is to express the degree of linear relationship between the two variables. The coefficient of correlation between two variables$x$and$y$is given by$r=\dfrac{\sigma^2x+\sigma^2y-y}{2\sigma_{x}\sigma_{y}}$where$\sigma_x$is the standard deviation of$x$and$\sigma_y$is the standard deviation of$y$The sum of the difference between the actual values of$Y$and its values obtained from the fitted regression line is always: 1. Zero 2. Positive 3. Negative 4. Minimum Correct Option: A Explanation: Let the actual values be$y_1,y_2,...,y_n$Let the values obtained from the fitted regression analysis be$\hat y_1,\hat y_2,...,\hat y_n$Since, the values are obtained from the fitted regression line. Therefore, the actual and obtained values are almost equal. Therefore,$y_1=\hat y_1$,$y_2=\hat y_2$, ... ,$y_n=\hat y_n$sum of the difference between the actual and obtained values is$(y_1-\hat y_1)+(y_2-\hat y_2)+...+(y_n-\hat y_n)=0$If all the actual and estimated values of$Y$are same on the regression line, the sum of squares of error will be: 1. Zero 2. Minimum 3. Maximum 4. Unknown Correct Option: A Explanation: Let the actual values be$y_1,y_2,...,y_n$Let the estimated values be$\hat y_1,\hat y_2,...,\hat y_n$Error is$actual-estimated$Given that actual and estimated values are equal. Therefore,$y_1=\hat y_1$,$y_2=\hat y_2$, ... ,$y_n=\hat y_n$sum of squares of error is$(y_1-\hat y_1)^2+(y_2-\hat y_2)^2+...+(y_n-\hat y_n)^2=0$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.$0$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:$x52143y584210$1.$y=-0.8x+7$2.$y=0.4x+7$3.$y=-0.4x+7$4.$y=-0.4x-7$Correct Option: C Explanation:$xyxyx^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. 1.$1$only 2.$2$only 3. Both$1$and$2$4. Neither$1$nor$2$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? 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$Correct Option: B Explanation: Line of regression of$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$1.$5.231$2.$4.591$3.$6.564$4.$8.231$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? 1. 1 only 2. 2 only 3. Both 1 and 2 4. Neither 1 nor 2 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$xy$A.M$18100$Standard Deviation$1420$1.$145.32$2.$44.23$3.$159.43$4.$561.12$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:$x246810y65432$1.$x+ 2y=14$2.$x-2y=14$3.$2x+ y=14$4.$2x- 2y=14$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=1b=$4$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. 1.$12$and$18$2.$18$and$12$3.$13$and$17$4.$17$and$13$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$x$. 1.$4x-y-11=0$2.$7x-3y-18=0$3.$4x-y-2=0$4.$7x-3y-4=0$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$x$. What would be the estimated sale on the advertisement expenditure of Rs$15$lakhs,on the basis of following data obtained from the company?.The coefficient of correlation is$0.8$. Advertising expenditure(in Rs.lakhs)$x$Sale (in Rs lakhs)$y$Mean$2090$standard Deviation$512$1. Rs$105$lakhs 2. Rs$106$lakhs 3. Rs$110$lakhs 4. Rs$120$lakhs Correct Option: B Explanation:$\bar x=20$,$\bar y=90$,$\sigma_x=5$,$\sigma_y=12$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$y-90=\dfrac{0.8\times 12}{5}(x-20)y-90=1.92(x-20)y=1.92x+90-38.4$Now, Subsitute the value of x=15, we get$y=1.92\times 15+51.6=80.4$Find the equation of$y$on$x$for the following data$x86475y98562$1.$y=2x -1.2$2.$y=1.2x +1.2$3.$y=1.2x -1.2$4.$y=1.2x -2$Correct Option: C Explanation:$xyx^2x\times y896472683648451620764942522510\sum x=30\sum y=30\sum x^2=190\sum x\times y=192$So,$\bar x=\dfrac{\sum x}{n}=\dfrac{30}{5}=6\bar y=\dfrac{\sum y}{n}=\dfrac{30}{5}=6b_{yx}=\dfrac{\sum xy-\dfrac{\sum x \times \sum y}{n}}{\sum x^2-\dfrac{(\sum
x)^2}{n}}=\dfrac{192-180}{190-180}=\dfrac{12}{10}=1.2$Regression line of y on x will be$y-\bar y=b_{yx}(x-\bar x)y-6=1.2(x-6)y=1.2x-7.2+6y=1.2x-1.2$For the variables$x$and$y$, the regression equations are given as$7x-3y-18=0$and$4x-y-11=0$. Find the arithmetic means of$x$and$y$respectively. 1.$3$and$1$2.$1$and$3$3.$2$and$4$4.$4$and$2$Correct Option: A Explanation: Since the two lines of regression interest at the point$(\bar{X}, \bar{Y} )$Replace$x$and$y$by$\bar{X}$and$\bar{Y}$respectively in the given regression equations. We get,$7\bar{X}- 3\bar{Y} - 18 = 0$and$4\bar{X}-\bar{Y} - 11 = 0$Solving these equations, we get$\bar{X} = 3$and$\bar{Y} = 1$Thus the arithmetic mean of$x$and$y$is given by$3$and$1$respectively. The two lines of regression are$x+2y-5=0$and$x+3y-8=0$. The coefficient of correlation between$x$and$y$is 1.$-0.72$2.$0.72$3.$-0.82$4.$0.82$Correct Option: C Explanation: Given two lines$x+2y-5=0, x+3y-8=0$. Consider$x+2y-5=0\Rightarrow x=-2y+5\Rightarrow r_1=-2$Consider$x+3y-8=0\Rightarrow y=-\dfrac{1}{3}x+\dfrac{8}{3}\Rightarrow r_2=-\dfrac{1}{3}$We know that$r^2=r_1 \times r_2\Rightarrow r^2=-2 \times -\dfrac{1}{3}\Rightarrow r^2=\dfrac{2}{3}\Rightarrow r=\pm \sqrt{\dfrac{2}{3}}$We know that, If both regression coefficients are negative,$r$would be negative.$\Rightarrow r=-\sqrt{\dfrac{2}{3}}=-0.82\$

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