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LINEAR CORRELATION · Gujarat Board (GSEB) STD 12 Commerce (GSEB - English Medium). 44 questions.

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Question 11 Mark
If $r(x,y):0.3$ then find $r(u,v)$ for the following.
$u=\frac{20-x}{3}$ and $v=\frac{20-y}{7}$
Answer
While defining $u$ and $v$ from the properties $($no. $4$ and no. $5),$ the value of $r(u, v)$ will be dependent on the signs of the coefficients of $X$ and $Y.$ i.e. $r(u, v) = r(x, y)$ or $-r(x, y)$
$r\left(\frac{20-x}{3}, \frac{20-y}{7}\right)=r(u, v)=0.8$
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Question 21 Mark
If $r(x,y):0.3$ then find $r(u,v)$ for the following.
$u=\frac{5-x}{2}$ and $v=\frac{5+y}{2}$
Answer
While defining $u$ and $v$ from the properties $($no. $4$ and no. $5),$ the value of $r(u, v)$ will be dependent on the signs of the coefficients of $X$ and $Y.$ i.e. $r(u, v) = r(x, y)$ or $-r(x, y)$
$r\left(\frac{5-x}{2}, \frac{5+y}{2}\right)=r(u, v)=-0.8$
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Question 31 Mark
If $r(x,y):0.3$ then find $r(u,v)$ for the following.
$u=\frac{2 x-3}{10}$ and $v=\frac{10-y}{100}$
Answer
While defining $u$ and $v$ from the properties $($no. $4$ and no. $5),$ the value of $r(u, v)$ will be dependent on the signs of the coefficients of $X$ and $Y$ .i.e. $ r(u, v) =r(x, y)$ or $-r(x, y)$
$r\left(\frac{2 x-3}{10}, \frac{10-y}{100}\right)=r(u, v)=-0.8$
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Question 41 Mark
If $r(x,y):0.3$ then find $r(u,v)$ for the following.
$u=\frac{x-5}{3}$ and $v=2 y+7$
Answer
While defining $u$ and $v$ from the properties $($no. $4$ and no.$ 5),$ the value of $r(u, v)$ will be dependent on the signs of the coefficients of $X$ and $Y$ .i.e. $r(u, v) = r(x, y)$ or $-r(x, y)$
$r\left(\frac{x-5}{3}, 2 y+7\right)=r(u, v)=0.8$
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Question 51 Mark
If $r(x,y):0.3$ then find $r(u,v)$ for the following.
$u = 1-10$ and $v = y+10$
Answer
While defining u and v from the properties $($no. $4$ and no. $5),$ the value of $r(u, v)$ will be dependent on the signs of the coefficients of $X$ and $Y.$
i.e. $r(u, v) = r(x, y)$ or $-r(x, y)$
$r(x - 10,y + 10) = r(u,v) = 0.8$
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Question 61 Mark
What is the main limitation of scatter diagram method ?
Answer
The main limitation of scatter diagram method is that we cannot obtain the exact measure of strength of correlation.
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Question 71 Mark
The correlation coefficient between $X$ and $Y$ is $0.4.$ What will be the value of correlation coefficient if $5$ is added in each observation of $X$ and $10$ is subtracted from each observation of $Y\ ?$
Answer
The correlation coefficient between $X$ and $Y$ will be $0.4$ because correlation coefficient does not change by the change of origin.
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Question 81 Mark
What can be said about the correlation between the annual import of crude oil and the number of marriages during the same period of time ?
Answer
There is no correlation between the annual import of crude oil and the number of marriages during the same period of time. Since both the variables are independent.
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Question 91 Mark
Altitude and the amount of oxygen in air.
Answer
There is negative correlation between altitude and the amount of oxygen in air. As height of altitude increases the amount of oxygen decreases.
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Question 101 Mark
The rate of inflation and the purchase power of a common man of a country when income of the common man is stable .
Answer
There is a negative correlation between rate of inflation and the purchase power of a common man of a country when income of the common man is stable .As the rate of the inflation increases , the Purchase power of a common man decreases.
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Question 111 Mark
The sales and profits of last five years for a mostly accepted product of the company.
Answer
There is positive correlation between sales and profit. As the sales increases , the profit also increases
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Question 121 Mark
The age of an adult person and life insurance premium at the time of taking insurance under a plan.
Answer
There is positive correlation between the age of an adult person and life insurance premium. As the age of person increases, the life insurance premium also increases.
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Question 131 Mark
Define correlation coefficient.
Answer
  • The measure of the strength of the linear correlation between two variables is called correlation coefficient. It is denoted by $r.$ Its value lies between $-1$ and $1$
  • Note: Identify whether there is a positive correlation or negative correlation between the following pairs of variables $($question $3$ to question $6)$
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Question 141 Mark
What will be the sign of $r$ if the value of co-variance is negative ?
Answer
Sign of correlation coefficient is dependent on co-variance of $X$ and $Y.$ Therefore the sign of $r$ will be negative.
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Question 151 Mark
If the value of $n(n^2-1)$ is six times the value $\sum d^2$ then what is the value of $r\ ?$
Answer
$n\left(n^2-1\right)=6 \Sigma d^2$
Now, $r=1-\frac{6 \Sigma d^2}{n\left(n^2-1\right)}$
Putting $n\left(n^2-1\right)=6 \Sigma d^2$
$r=1-\frac{6 \Sigma d^2}{6 \Sigma d^2}$
$\therefore r=1-1=0$
Hence, the value of $r$ is $' 0 '.$
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Question 161 Mark
Define correlation.
Answer
If there are simultaneous changes in the values of two variables due to direct or indirect cause effect then it is said that there is correlation between two variables.
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Question 171 Mark
Which is the axiom to be taken from the very beginning to study correlation?
Answer
To study the correlation the axiom to be taken from the very beginning is that “there is cause and effect relationship between two variables.”
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Question 181 Mark
State the names of methods of obtaining measure of correlation.
Answer
The two methods to obtain the measure of correlation are as follows: $(1)$ Karl Pearsons method and $(2)$ Spearman’s rank correlation method.
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Question 191 Mark
What is cause and effect relationship?
Answer
Out of two related variables, due to the cause if there is increase or decrease in the values of the one variable and the result is increase or decrease in the value of another variable, then we can say that there exists cause and effect relationship between two variables.
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Question 201 Mark
Which type of correlation is between Demand and Price?
Answer
There is partial negative linear correlation between Demand and Price.
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Question 211 Mark
If there is negative correlation between two variables $X$ and $Y,$ what will be the sign of covariance $(X, Y)?$
Answer
If there is negative correlation between the variables $X$ and $Y,$ then the sign of covariance $(X, Y)$ will be negative.
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Question 221 Mark
Which is the method used to know the nature of correlation between two variables?
Answer
The scatter diagram method Is used to know the nature of correlation between two variables.
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Question 231 Mark
Define : Perfect negative linear correlation
Answer
When all the points on a scatter diagram are on a straight line going downwards from upper left-hand corner to lower right-hand corner of the graph paper, then the variables $X$ and $Y$ are said to have perfect negative linear correlation between them.
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Question 241 Mark
Define perfect positive linear correlation.
Answer
When all the points on a scatter diagram are on a straight line going upwards from lower left-hand corner to upper right-hand corner of the graph paper, then the variables $X$ and $Y$ are said to have perfect positive linear correlation between them.
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Question 251 Mark
Give two practical illustrations of negative correlation.
Answer
The illustrations of negatively correlated variables are as follows: $(1)$ ‘Cost of living expenditure’ and ‘Savings’ of families. $(2)$ ‘Population’ and ‘Per capita income’ at different periods of time in a region.
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Question 261 Mark
Give two practical illustrations of positive correlation.
Answer
The illustrations of positively correlated variables are as follows: $(1)$ ‘Consumption of fertilizers’ and ‘Yield of grain.’ $(2)$ ‘Monthly income’ and ‘Monthly expenditure’ of families.
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Question 271 Mark
Give two practical illustrations of two correlated variables.
Answer
The practical illustrations of two correlated variables are as follows: $(1)$ ‘The cost of advertisement’ and ‘Sale’ of a company. $(2)$ ‘The price’ and ‘demand’ of a commodity during a interval of time.
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Question 281 Mark
Which statistician had first suggested the measure of correlation?
Answer
The English statistician Karl Pearson had first suggested the measure of correlation.
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Question 291 Mark
Write the definition of scatter diagram.
Answer
Plotting the values of variable $X$ on $X-$axis and the values of variable $Y$ on $Y-$axis, by taking proper scale on a graph paper, the graph showing the plotted points corresponding to a pairs $(x_1, y_1), (x_2, y_2) (x, y)$ is called a scatter diagram.
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Question 301 Mark
The following information is obtained to study the relationship between the advertisement cost and the sales of electric fans of the companies manufacturing electric fans. Find the correlation coefficient advertisement cost and the sales by Karl Pearson’s method.
Company $A$ $B$ $C$ $D$ $E$ $F$
Advertisement Cost $($lakh $Rs.)$ $140$ $120$ $80$ $100$ $80$ $180$
Sales of electric fans $($Crore $Rs.)$ $35$ $45$ $15$ $40$ $20$ $50$
Answer
Ans: Here, Let $X =$ Advertisement cost and $Y =$ Sales of electric fans $($Crore $Rs.).$
Since means $\bar{x}$ and $\bar{y}$ of variables is in the fraction, we prepare the calculative table as follows :
Company $X$ $Y$ $u =\frac{y-A}{C_x}
A = 550, C_x = 50$		 $v =\frac{y-B}{C_y}
B = 25, C_y = 5$		 $uv$ $u^2$ $v^2$
$A$ $140$ $35$ $3$ $1$ $3$ $9$ $1$
$B$ $120$ $45$ $1$ $3$ $3$ $1$ $9$
$C$ $80$ $15$ $-3$ $-3$ $9$ $9$ $9$
$D$ $100$ $40$ $-1$ $2$ $-2$ $1$ $4$
$E$ $80$ $20$ $-3$ $-2$ $6$ $9$ $4$
$F$ $180$ $50$ $7$ $4$ $28$ $49$ $16$
Total $\Sigma{x} = 700$ $\Sigma{y} = 205$ $\Sigma{u} = 4$ $\Sigma{v} = 5$ $\Sigma{uv} = 47$ $\Sigma{u^2}
= 78$		 $\Sigma{v^2}
= 43$
$\bar{x} {=} \frac{\sum \mathrm{x}}{\mathrm{n}}=\frac{700}{6}= 116.67$ and $\bar{y} {=}\frac{\sum y}{n}=\frac{205}{6}= 34.17$
$r = \frac{n \sum u v-\left(\sum u\right)\left(\sum v\right)}{\sqrt{n \sum u^2-\left(\sum u\right)^2} \times \sqrt{n \sum v^2-\left(\sum v\right)^2}} =$
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Question 311 Mark
What is correlation coefficient ?
Answer
Correlation co-efficient is a numerical measurement which shown strength or degree between two variable which are linear it is denoted by $‘r’.$
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Question 321 Mark
Give the two examples of Negative correlation.
Answer
Following examples are shown negative correlation between variables.
  • Price of goods and its demand,
  • Living cost of family and saving,
  • Propertion of population at different times of a state and income per head,
  • Minimum temperature of a day in winter and sales of woollen cloth,
  • Height of a place from sea level and proportion Of Oxygen in air at place.
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Question 331 Mark
Give two examples of positive correlation.
Answer
Following examples are there for positive correlation between variables :
  • Price of goods and supply.
  • Income of person and expense,
  • Advertisement expense and sale.
  • Use of fertility and production of grain,
  • Use of Electricity and its bill,
  • No. of vehicles and No. of accident.
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Question 341 Mark
Give two examples of positive correlation.
Answer
Following examples are there for positive correlation between variables :
  • Price of goods and supply.
  • Income of person and expense,
  • Advertisement expense and sale.
  • Use of fertility and production of grain,
  • Use of Electricity and its bill,
  • No. of vehicles and No. of accident.
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Question 351 Mark
When it will be said that variables are “There is a correlation between two variables ?
Answer
If there is a fluctuation the value related random variable together, on because of absolute or relative cause and effect relationship, then it is said that ‘There is a correlation between two variable.’
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Question 361 Mark
How many types of correlation ?
Answer
There are mainly two types of correlation $(i)$ Positive correlation and $(ii)$ Negative Correlation.
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Question 371 Mark
What is meant by linear correlation ?
Answer
If all the value of set of two variable shown on graph and all the point are in strength line, then it is linear correlation.
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Question 381 Mark
What is meant by cause effect relation ?
Answer
If there are two correlated random variable, and if the change in the values of one variable $($cause$)$ is due to the change in the value of other variable $($effect$)$ then it is cause and effect relation between two correlated random variable.
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Question 391 Mark
Which prior assumption is made in the study of correlation ?
Answer
In the begining of the study correlation between two variables, we make assumption that there is relation of cause and effect between two variables.
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Question 401 Mark
What is meant by correlate random variable ?
Answer
There is a correlation between two random variable, it is correlated random variable.
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Question 411 Mark
What is the main limitation of scatter diagram method ?
Answer
The main limitation of the scatter diagram method is that it shows the nature of the relationship but does not provide a specific numerical value (measure) for the degree of correlation.
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Question 421 Mark
What will be the sign of 'r' if the value of the covariance is negative ?
Answer
If the value of the covariance is negative, then the sign of correlation coefficient 'r' will also be negative.
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Question 431 Mark
What will be the sign of correlation r if the value of covariance is negative ?
Answer
If the value of covariance is negative, the sign of the correlation coefficient $ r $ will also be negative.
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Question 441 Mark
Define correlation coefficient.
Answer
A measure of the strength of linear relationship between two variables is called correlation coefficient. It is denoted by 'r'.
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