Question 12 Marks
if the correlation coefficient between two variables $X$ and $Y$ is $0.5,$ find the value of the following : $(i)\ r (x, - y)\ (ii)\ r (- x, y)\ (iii)\ r (- x, - y)$
AnswerHere, $r(x, y)=0.5$ From the property no. $5$ of correlation coefficient,
$(i)\ r(x,-y)=-r(x, y)=-0.5$
$(ii)\ r(-x, y)=-r(x, y)=-0.5$
$(iii)\ r(-x,-y)=r(x, y)=0.5$
View full question & answer→Question 22 Marks
The following results are obtained from a bivariate data.
$n=10, \Sigma(x-\bar{x})(y-\bar{y})=72, s_{x}=3$ and $\Sigma(y-\bar{y})^{2}=160$ Find the correlation coefficient.
AnswerFrom the available results, first we shall find $s_{y}$.
$s_{y}=\sqrt{\frac{\Sigma(y-\bar{y})^{2}}{n}}=\sqrt{\frac{160}{10}}=\sqrt{16}=4$
Now, substituting the necessary values in the following formula,
$ r=\frac{\Sigma(x-\bar{x})(y-\bar{y})}{n s_{x} s_{y}}$
$ =\frac{72}{10(3)(4)}$
$ \quad=\frac{72}{120}$
$ \therefore r =0.6$
View full question & answer→Question 32 Marks
Determine the value of the correlation coefficient from the following results.
$\operatorname{Cov}(x, y): s_{x}^{2}=3: 5$ and $s_{x}: s_{y}=1: 2$
AnswerHere, $\operatorname{Cov}(x, y): s_{x}{ }^{2}=3: 5$
$\therefore \frac{\operatorname{Cov}(x, y)}{s_{x}{ }^{2}}=\frac{3}{5}$ and $s_{x}: s_{y}=1: 2$
$ \therefore \frac{s_{x}}{s_{y}}=\frac{1}{2}$
Now, $ r=\frac{\operatorname{Cov}(x, y)}{s_{x} s_{y}}=\frac{\operatorname{Cov}(x, y)}{s_{x}{ }^{2}} \times \frac{s_{x}}{s_{y}}$
$=\frac{3}{5} \times \frac{1}{2}$ $=\frac{3}{10}$
$\therefore r=0.3$
View full question & answer→Question 42 Marks
A transport company wants to know the relation between driving experience and the number of accidents by the drivers. The sum of squares of differences in the ranks given to driving experience and the number of accidents by eight drivers is found to be $126.$ Find the rank correlation coefficient.
AnswerHere, $n=8$ and the sum of squares of difference in the ranks is $126 $, i.e. $\Sigma d^{2}=126$.
$r =1-\frac{6 \Sigma d^{2}}{n\left(n^{2}-1\right)}$
$ =1-\frac{6(126)}{8(64-1)}$
$ =1-\frac{756}{504}$
$ =1-1.5$
$r =-0.5$
View full question & answer→Question 52 Marks
To study the relationship between the marks obtained in Statistics $(X)$ and marks in Economics $(Y)$ of the students of a school, a sample of ten students is taken and the following information is obtained.
$\Sigma(x-\bar{x})(y-\bar{y})=120, \Sigma(x-\bar{x})^{2}=80, \Sigma(y-\bar{y})^{2}=500$
Find the value of $r.$
AnswerHere, $n=10$ and the following formula is suitable to the given data.
$ r =\frac{\Sigma(x-\bar{x})(y-\bar{y})}{\sqrt{\Sigma(x-\bar{x})^{2}} \cdot \sqrt{\Sigma(y-\bar{y})^{2}}}$
$ =\frac{120}{\sqrt{80} \cdot \sqrt{500}}$
$ =\frac{120}{\sqrt{40000}}$
$ =\frac{120}{200}$
$\therefore r =0.6 $
View full question & answer→Question 62 Marks
When is it necessary to use the rank correlation ?
Answer
- Spearman’s rank correlation is used in the following cased
- $(a)$ When given data cannot be represented in the numerical from, i.e when ranks are given according to their magnitudes.
- $(b)$ When dispersion is more or when the extreme observation are present in the data of two correlated variables.
View full question & answer→Question 72 Marks
Explain : Perfect negative correlation.
Answer
- If all the points of the scatter diagram obtained by plotting n ordered pairs of observations of two correlated variables $x$ and $y,$ lie on one line which goes in downward direction from left to right, then we can say that there is perfect negative correlation between variable $x$ and $y.$
- Here values of both the variables change in the opposite direction and increase or decrease in the same proportion.
- This type of correlation can be expressed by the equation $y = a + bx(b < 0)$
View full question & answer→Question 82 Marks
Explain : Perfect positive correlation.
Answer
- If all the points of the scatter diagram obtained by plotting $n$ ordered pairs of observations of two correlated variables $x$ and $y,$ lie on one line which goes in upward direction from left to right, then we can say that there is perfect positive correlation between variable $x$ and $y.$
- Here values of both the variables change in the same direction and increase or decrease in the same proportion.
- This type of correlation can be expressed by the equation $y = a + bx (b > 0).$
View full question & answer→Question 92 Marks
What is spurious correlation ?
Answer
- When there are simultaneous changes in two variables and there is cause effect relation between two variables then this type of relation is known as correlation.
- In many cases there is no meaningful relation but value of correlation coefficient r calculated on the basis of observations of two variables comes very near to $1.$
- For example there is no meaningful relation between America's per capita income and number of deaths caused due to cancer in India but still if we calculate correlation coefficient r between them by taking observations during a particular time, its value will be near $1.$
- This type of correlation is known as spurious correlation.
View full question & answer→Question 102 Marks
Define : Scatter Diagram.
Answer
- By taking values of variable $X$ on $x$-axis and corresponding values of $Y$ on $y$-axis with appropriate scale and plotting n sample points $\left(\mathrm{x}_1, \mathrm{y}_1\right),\left(\mathrm{x}_2, \mathrm{y}_2\right) \ldots \ldots\left(\mathrm{x}_n, \mathrm{y}_n\right)$ on the graph paper, a diagram is obtained which is called scatter diagram.
- From the pattern of points on the scatter diagram, nature of correlation and strength of correlation can be known to some extent.
View full question & answer→Question 112 Marks
Write the assumptions of Karl Pearson's method.
Answer
- Karl Pearson's correlation coefficient is based on the following assumptions.
- $(a)$ There is a linear relationship between two variables.
- $(b)$ There is a cause effect relationship between two variables.
- Correlation is meaningless if there is no such relationship.
View full question & answer→Question 122 Marks
Explain the meaning of negative correlation with illustration.
Answer
- For two correlated variables, if the value of one variable increases then the value of other variable decrease means both the variables increase and decrease in the opposite direction and there is a cause effect relation between the two variables then we can say there is negative correlation between the two variables.
- Person's expenditure and savings, the minimum temperature in a day during winter and sale of woollen clothes, price of a commodity and its demand, elevation from sea level and amount of oxygen in air etc are the examples of negative correlation.
- There is a negative correlation between price of a commodity and its demand in the following example.
- When there is increase in price of commodity, there is a decrease in its demand.
| Price of commodity$($in $Rs.)$ |
$10$ |
$12$ |
$15$ |
$20$ |
$25$ |
$30$ |
$45$ |
| Demand of commodity $($in units$)$ |
$130$ |
$120$ |
$100$ |
$95$ |
$90$ |
$84$ |
$80$ |
View full question & answer→Question 132 Marks
For $10$ the pairs of the observations,
$\sum \mathbf{d}^2 = 120.$ Find the value of rank correlation coefficient.
AnswerHere, $\mathrm{n}=10 ; \Sigma \mathrm{d}^2=120$
Now, $r=1-\frac{6 \Sigma d^2}{n\left(n^2-1\right)}$
$ =1-\frac{6(120)}{10(100-1)}$
$ =1-\frac{720}{10 \times 99}$
$ =1-\frac{720}{990}$
$ =1-0.73$
$ =0.27$
$ \therefore r=0.27$
View full question & answer→Question 142 Marks
Find the value of $r$ if $\Sigma(x-\bar{x})(y-\bar{y})=-65, S_x=3, S_y=4$ and $n=10$.
AnswerHere $\Sigma(x-\bar{x})(y-\bar{y})=-65, S_x=3, S_y=4$ and $n=10$.
Now, $\mathrm{r}=\frac{\Sigma(x-\bar{x})(y-\bar{y})}{n \cdot S_x \cdot S_y}$
$=\frac{-65}{10 \times 3 \times 4}$
$ =\frac{-65}{120}$
$ =-0.54$
View full question & answer→Question 152 Marks
Find the value of $r$ if $\operatorname{Cov}(x, y)=120, S_x=12$ and $S y=15$.
AnswerHere, $\operatorname{Cov}(x, y)=120, S_x=12, S_y=15$
Now, $\mathrm{r}=\frac{\operatorname{Cov}(x, y)}{\mathrm{S}_x \cdot \mathrm{S}_y}=\frac{120}{12 \times 15}=\frac{120}{180}=0.67$
View full question & answer→Question 162 Marks
In which situation, the values of Karl Pearson’s correlation coefficient and spearman’s rank correlation coefficient are equal ?
AnswerWhen the values of two variables $X$ and $Y$ are some arrangement of first n natural numbers, the correlation coefficient obtained by Karl Pearson’s method and Spearman’s method are equal.
View full question & answer→Question 172 Marks
Explain the meaning of positive correlation with illustration.
Answer
- For two correlated variables, if the value of one variable increases then the value of other variable also increases.
- And if the value of one variable decreases, the value of another variable also decreases means both the variables increase or decrease in the same direction and there is cause effect relation between the two variables then we can say that there is a positive correlation between the two variables for e.g. the age of husband, number of vehicles and the number of accidents, advertisement cost and sale, price and demand, income and expenditure , sales and profit are the examples of positive correlation.
- There is a positive correlation between monthly income and monthly expenditure in the following example.
- When there is increase in monthly income, there is an increase in monthly expenditure also.
-
| Monthly Income |
$2500$ |
$2800$ |
$3200$ |
$3600$ |
$4200$ |
$4500$ |
$4800$ |
| Monthly Expenditure |
$1500$ |
$2000$ |
$2800$ |
$3000$ |
$3800$ |
$4000$ |
$4400$ |
View full question & answer→Question 182 Marks
Find the correlation coefficient from the following information of rainfall $(X) ($in cm$ )$ and yield $(Y) ($tons per hactare$)$ for the last $10$ years of a district: $n=10, \operatorname{Coy}(x, y)=30$, S.D. of $X=5$ and variance of $Y=144$
AnswerHere, $\mathrm{n}=10 ; \operatorname{Cov}(\mathrm{x}, \mathrm{y})=30 ; \mathrm{S}_{\mathrm{x}}=5 ; \mathrm{S}_{\mathrm{y}}{ }^2=144 \operatorname{Cov}(\mathrm{x}, \mathrm{y})$
$\therefore \mathrm{S}_{\mathrm{y}}=12$
Now, $r=\frac{\operatorname{Cov}(x, y)}{S_x \cdot S_y}$
$=\frac{30}{5 \times 12}=\frac{30}{60}=0.5$
Hence, the correlation coefficient between rainfall and yield of crop obtained is $0.5 .$
View full question & answer→Question 192 Marks
If $r =-0.55, \sum d^{2}=341$; find $n$
View full question & answer→Question 202 Marks
If $n=7, \sum d^{2}=4, CF =1$; Find $r$
View full question & answer→Question 212 Marks
If $r =-0.5, n =10$ find $\sum d^{2}$
View full question & answer→Question 222 Marks
For two variables $X$ and $Y$, we get $n\left(n^{2}-1\right)=5 \geq d^{2}=720$ Calculate the coefficient of rank correlation.
View full question & answer→Question 232 Marks
If $n =7,2 d^{2}=4, CF =1 ;$ Find $r$
View full question & answer→Question 242 Marks
If $r =-0.5, n =10$ find $\sum d^{2}$
View full question & answer→Question 252 Marks
For two variables $X$ and $Y$, we get $n\left(n^{2}-1\right)=5 \sum d^{2}=720$ Calculate the coefficient of rank correlation.
View full question & answer→Question 262 Marks
Ranks are given to $11$ students of standard $12$ according to their marks in Statistics and marks in Accountancy. The differences of respective ranks are $1, -1, 2, 1, 0, -2, -3, 1, 2, -1$ and $0.$ Find the coefficient of rank correlation between the marks in Statistics and marks in Accountancy.
View full question & answer→Question 272 Marks
Variables $u$ and $v$ are defined as $u=\frac{x}{10} 20$ collected from a sample of size $15 .$ Calculate $r(u, v)$ and from that find $r(x, y)$. $\sum u-12, \sum v-15, \sum u^{2}-15, \sum v^{2}-30, \sum u v=20$
View full question & answer→Question 282 Marks
Variables $u$ and $v$ are defined as $u=x-18$ and $v=y-22$. For a sample of size $10 , \sum(x-18)=-8, \sum(y-22)=-6, \sum-6,(x-18) 2=26,(x-18)^{2}=$ $26, \sum(y-22)^{2}=18$ and $\sum(x-18)(y-22)=20$ calculate $r(u, v)$ and from that find $r(x, y)$.
View full question & answer→Question 292 Marks
From the sample survey of $10$ observations each of variables $X$ and $Y$ the following results are obtained. Find the coefficient of correlation between these two variables. $ \sum x=5, \sum y=5, \sum x^{2}=305, \sum y^{2}=25 \text { and } \sum x y=-65 $
View full question & answer→Question 302 Marks
Following results are obtained from the data on marks of Economics $(X)$ and Statistics $(Y)$ obtained by $7$ students of standard $12: \sum x=70, \sum y=63, \sum x^{2}=728, \sum y^{2}=651$ and $\sum x y=676$ Find $r(x, y)$.
View full question & answer→Question 312 Marks
Following information was collected from a survey of $9$ students regarding their marks in Accountancy $(X)$ and their marks in Statistics $(Y)$ in $10$ mark test in each subject: $\sum(x-2)=20, \sum(y-3)=25, \sum(x-3)(y-4)=46, \sum(x-5)^{2}=$ $53, \sum(y+3)^{2}=721$ Find the coefficient of correlation between $X$ and $Y$.
View full question & answer→Question 322 Marks
Following data is collected for two variables $X$ and $Y$ : $, n=10, \sum\left(x_{i}-40\right)=0, \sum\left(y_{i}-56\right)=0, \sum\left(x_{i}-40\right)^{2}=576, \sum\left(y_{i}\right.$ $56)^{2}-900, \sum\left(x_{i}-40\right)\left(y_{i}-56\right)--455$ Find the coefficient of correlation between $X$ and $Y$ from the data given above.
View full question & answer→Question 332 Marks
Find coefficient of correlation between $X$ and $Y$ from the following data: $ n=15, \Sigma x=975, \Sigma y=1530, \Sigma(x-65)^{2}=576, \Sigma(y-102)^{2}=1225$
$ \text { and } \sum(x-65)(y-102)=714 $
View full question & answer→Question 342 Marks
From $12$ pairs of observations on variables $X$ and $Y$, following data is collected: $\bar{x}=252, \bar{y}=108, \sum(x-252)^{2}=1024, \sum(y-108)^{2}=324$ and $\sum(x-$ $252)(y-108)=-432$ Find $r(x, y)$
View full question & answer→Question 352 Marks
Find the coefficient of correlation from the following: $n=10, \sum(x-\bar{x})(y-\bar{y})=120, \sum(x-\bar{x})^{2}=90, S_{y}=2.5$,
Answer$S_x = 3 , r = 0.5$
View full question & answer→Question 362 Marks
Following information was collected from $10$ pairs of observations of two interdependent variables $X$ and $Y$ : $ \begin{aligned} &\sum(x-52.5)=0, \sum(y-120.5)=0, \\ &\sum(z-52.5)^{2} \\ &\sum(x-52.5)(y-120.5)=180 \end{aligned} $ Find the coefficient of correlation between $X$ and $Y$.
View full question & answer→Question 372 Marks
For a data of variables $X$ and $Y n=20$, coefficient of correlation $=-0.5, S_{y}=5$ and $\sum(x-\bar{x})(y-\bar{y})=-150$, Find $S_{x}^{2}$.
View full question & answer→Question 382 Marks
If $n-15, S_{z}-4.5, S_{y}-6.2, \sum(x-\bar{x})(y-\bar{y})--362$, find $r(x, y)$
View full question & answer→Question 392 Marks
For a sample of size $10$ of two interdependent random variables $X$ and $Y$, $\operatorname{Cov}(x, y): S_{y}^{2}-1: 2$ and $S_{z} S_{y}-3: 4$ Find $r(x, y)$
View full question & answer→Question 402 Marks
For a sample of size $20$ of two variables $r(x, y)=0.6, \operatorname{Cov}(x, y)=60, S_{y}=12$ Find $S_{x}$
View full question & answer→Question 412 Marks
The coefficient of correlation between variables $X$ and $Y$ is $-0.8$. Covariance of $X$ and $Y$ is $-120$ and variance of $X$ is $100.$ Find standard deviation of $Y$.
View full question & answer→Question 422 Marks
From a sample of size $10$ of varlables $X$ and $Y$, we get $\frac{1}{3} \operatorname{Cov}(x, y)=3 S_{x}=$ $5 S_{y}=60$ Calculate the value of $r(x, y)$. If $u=2 x+3$ and $v=\frac{y-20}{4}$, what will be the value of $r(u, v)$ ?
View full question & answer→Question 432 Marks
From $12$ pairs of observations of variables $X$ and $Y$, we get Cov $(x, y)=$ $-240, S_{x}=15$ and $S_{y}=20$ calculate r( $(x, y)$.
View full question & answer→Question 442 Marks
If the correlation coefficient between two variables X and Y is 0.8 find the value of the following :
(i) $r(x,-y)$
(ii) $r(-x,-y)$
Answer(i) $r(x,-y) = -0.8$ (Change in sign of one variable changes the sign of r)
(ii) $r(-x,-y) = 0.8$ (Change in sign of both variables keeps r same)
View full question & answer→Question 452 Marks
Write the properties of correlation coefficient.
Answer1. The value of correlation coefficient 'r' lies between -1 and 1, i.e., $-1 \le r \le 1$.
2. It is a unit-less measure.
3. Correlation coefficient between X and Y is the same as between Y and X, i.e., $r(x,y) = r(y,x)$.
4. It is independent of change of origin and scale.
View full question & answer→Question 462 Marks
Find the value of r if $ CoV(x,y)=120 $, $ S_{x}=12 $, $ S_{y}=15 $.
Answer$ r = \frac{Cov(x,y)}{S_{x} \cdot S_{y}} $
$ r = \frac{120}{12 \cdot 15} = \frac{120}{180} = \frac{2}{3} = 0.67 $
View full question & answer→Question 472 Marks
Explain the meaning of positive correlation with an illustration.
AnswerWhen changes in the values of two variables are in the same direction, the correlation between them is said to be positive.
Example : Relation between Price and Supply of a commodity.
View full question & answer→