6 Marks Question

Question 16 Marks

Following table gives the distribution of companies according to the size of capital. Using step deviation method, find out the mean size of the capital of a company.

Capital (Rs. in lakh)	Less than 5	Less than 10	Less than 15	Less than 20	Less than 25	Less than 30
Number of Companies	20	27	29	38	48	53

Answer

Capital (Rs. in lakh)	Number of Companies (f)	MidValue (m)	dim(m−A) (A =12.5)	$\begin{array}{l} d ^{\prime} m \left(\frac{ m }{ c }\right) ( c = 5 )\end{array}$	fd' m
0-5	20	2.5	-10	-2	$\left.\begin{array}{c}-40 \\ -7\end{array}\right]-47$
5-10	27-20=7	7.5	-5	-1	$\left.\begin{array}{c}-40 \\ -7\end{array}\right]-47$
10-15	29-27=2	12.5	0	0	0
15-20	38-29=9	17.5	+5	+1	$\left.\begin{array}{l}+9 \\ +20 \\ +15\end{array}\right]+44$
20-25	48-38=10	22.5	+10	+2
25-30	53-48=5	27.5	+15	+3
	$\Sigma f=53$				$\Sigma f d^{\prime} m=-3$

Here,
$\begin{array}{l}
A=12.5, \Sigma f=53, \Sigma f d^{\prime} m=-3 \text { and } c=5 \\
\bar{X}=A+\frac{\Sigma f d^{\prime} m}{\Sigma f} \times c=12.5+\frac{(-3)}{53} \times 5=R s .12 .5-0.28=R s .12 .22 \\
\therefore \bar{X}=12.22
\end{array}$
Hence, mean size of capital is Rs. 12.22 lakhs

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Question 26 Marks

Give formula for:
a. Simple mean in individual series by short cut method
b. Weighted mean
c. Simple mean in continuous series by direct method
d. Simple mean in discrete series by short cut method
e. Combined Mean
f. Simple mean in continuous series by step deviation method

Answer

(a) $\bar{x}=\frac{1}{n} A+\frac{\sum d}{N}$
(b) Weighted Mean $=\Sigma W X / \Sigma W$
(c) $\bar{x}=\frac{\sum_{i=1}^n f_i m_i}{\sum_{i=1}^n f_i}$
(d) $\bar{x}=A+\frac{\sum_{i=1}^n f_i d_i}{\sum_{i=1}^n f_i}$
(e) Combined Mean $\bar{x} 12=\frac{\bar{x}_1 N_1+\bar{x}_2 N_2}{N_1+N_2}$
(f) $\bar{x}=A+\frac{\sum d^{\prime}}{N} \times i$

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Question 36 Marks

From the following data, calculate coefficient of correlation between age and playing habits.

Age Group	20 -30	30 - 40	40 - 50	50 - 60	60 - 70
Number of students	25	60	40	20	20
Number of Regular Players	10	30	12	2	1

Answer

First, we are required to calculate the percentage of regular players.
Calculation of Percentage of Regular Players:

Number of Students	Number of Regular Players	Percentage of Regular Players
25	10	$\frac{10}{25} \times 100=40$
60	30	$\frac{30}{60} \times 100=50$
40	12	$\frac{12}{40} \times 100=30$
20	2	$\frac{2}{20} \times 100=10$
20	1	$\frac{1}{20} \times 100=5$

Denoting mid value of age as X and percentage of regular players as Y.

Age Group	X	dx(X - A), A = 45	$\begin{array}{c}d x^{\prime}\left(\frac{d z}{a}\right), c_1=10\end{array}$	$dx ^{\prime 2}$	Y	dy(Y - A), A = 30	$\begin{array}{c} dy ^{ \prime }\left(\frac{d y}{c_2}\right), c _2=5\end{array}$	$dy ^{\prime 2}$	dx'dy'
20 -30	25	-20	-2	4	40	10	2	4	-4
30-40	35	-10	-1	1	50	20	4	16	-4
40-50	45	0	0	0	30	0	0	0	0
50-60	55	10	1	1	10	-20	-4	16	-4
60 -70	65	20	2	4	5	-25	-5	25	-10
			$\Sigma d x^{\prime}=0$	\begin{array}{c}\Sigma d x^2=10\end{array}			$\Sigma d y^{\prime}=-3$	\begin{array}{c}\Sigma \Sigma d^{\prime 2}=61\end{array}	\begin{array}{c}\Sigma dx^{\prime} dy y^{\prime}=-22\end{array}

Here, $n =5, \Sigma dx ^{\prime}=0, \Sigma dx ^{\prime 2}=10, \Sigma dy ^{\prime}=-3, \Sigma dx ^{\prime} dy ^{\prime}=-22 \Sigma dy ^{\prime 2}=61$
Now,Putting the values in the given formula:
$\begin{array}{l}r=\frac{\Sigma d x^{\prime} d y^{\prime}-\frac{\Sigma d x^{\prime} \times \Sigma d y^{\prime}}{n}}{\sqrt{\Sigma d x^{\prime 2}-\frac{\left(\Sigma d x^{\prime}\right)^2}{n}} \times \sqrt{\Sigma d d y^2-\frac{\left(\Sigma d y^{\prime}\right)^2}{n}}}= \\ =\frac{-22-\frac{0 \times-3}{5}}{\sqrt{10-\frac{(0)^2}{5}} \times \sqrt{61-\frac{(-3)^2}{5}}} \\ =\frac{-22}{\sqrt{10} \times \sqrt{61-1.8}}=\frac{-22}{\sqrt{10} \times \sqrt{59.2}} \\ =\frac{-22}{3.16 \times 7.69}=\frac{-22}{24.3}=-0.90\end{array}$
- Therefore, Karl Pearson's coefficient of correlation between age and playing habits is $=0.90$.
- Interpretation of $r$
1. It indicates that there is a high degree of negative correlation between age and playing habits.
2. It indicates that as age increases, the tendency to play decreases.

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Economics 6 Marks Question

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