Case study (4 Marks)

Question 14 Marks

Read the text carefully and answer the questions:
Five friends Mohit, Sachin, Rohit, Mohan and kapil were playing in a ground, where they sit in a row in a straight line.

(a) In how many ways these five students can sit in a row?
(b) Total number of sitting arrangements if Mohit and Sachin sit together:
(c) What are the possible arrangements if Rohit and Mohan sits at the extrement positions?
(d) What are the possible orders if Kapil is sitting in the middle?

Answer

Read the text carefully and answer the questions:
Five friends Mohit, Sachin, Rohit, Mohan and kapil were playing in a ground, where they sit in a row in a straight line.

(i) Total number of ways $=5!=120$
(ii) Two position are fixed for Mohit and Sachin therefore considering it as one unit, total students left $=3+1=4$
Total possible arrangement $=4!\times 2!=48$
(iii)Total possible arrangements $=3!\times 2!=12$
(iv)Total possible arrangements $=4$ ! $=24$

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Question 24 Marks

Read the text carefully and answer the questions:
A mobile number is having 10 digits. It is not just a group of numbers strung out at random. All mobile numbers have 3 things in common, a 2-digit Access code (AC), a 3-digit provider code (PC), and a 5 digit subscriber code (SC). AC code and PC code are fixed, then
(a) How many mobile number are possible if number start with 98073 and no other digit can repeat.
(b) How many AC code are possible if both digit in AC code are different and must be greater than 6.
(c) How may mobile number are possible if AC and PC code are fixed and digits can repeat
(d) How many mobile numbers are possible with AC code 98 and PC code 123 and digit used in AC and PC code will not be used in SC code.

Answer

Read the text carefully and answer the questions:
A mobile number is having 10 digits. It is not just a group of numbers strung out at random. All mobile numbers have 3 things in common, a 2-digit Access code (AC), a 3-digit provider code (PC), and a 5 digit subscriber code (SC). AC code and PC code are fixed, then
(i) 98073 V IV III II I
The digits which can be used are, $6,5,4,2,1$
Number of ways to fill the 5 places
=5!=120
(ii) Digits which can be used in AC code are 7, 8, 9
Total AC code $={ }^{3} \mathrm{P}_{2}=3!=6$
(iii) If AC and PC are fixed then only 5 digits is to be filled if digits can repeat then total ways $=100000=10^{5}$
(iv)Total ways $=5 \times 5 \times 5 \times 5 \times 5=3125$

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Question 34 Marks

Read the text carefully and answer the questions:
In class of Statistics, teacher was discussing the concept of Measures of Correlation, in which he was discussing about Karl Pearson's Coefficient of Correlation. During his class, he discussed the following few points on this: This is the best method for finding correlation between two variables provided the relationship between the two variables is linear. This method is also known as product moment correlation coefficient. Pearson's correlation coefficient may be defined as the ratio of covariance between the two variables to the product of the standard deviations of the two variables.
If the two variables are denoted by x and y and of the corresponding bivariates data are $\left(\mathrm{x}_{\mathrm{i}}, \mathrm{y}_{\mathrm{i}}\right)$ for $\mathrm{i}=1,2,3, \ldots$, n , then the coefficent of correlation between x and y due to Karl Pearson, is given by:
$r=r_{x y}$
or, $\mathrm{r}_{\mathrm{xy}}=\frac{\operatorname{Cov}(x, y)}{\sqrt{\operatorname{Var} x} \cdot \sqrt{\operatorname{Var} y}}$
$=\frac{\operatorname{Cov}(x, y)}{\sigma_{x} \cdot \sigma_{y}}$
where,
$\operatorname{cov}(\mathrm{X}, \mathrm{y})=\frac{\Sigma(x-\bar{x})(y-\bar{y})}{N}$
$=\frac{\Sigma x y}{N}-\bar{x} \cdot \bar{y}$
$\sigma_{x}=\sqrt{\frac{\Sigma(x-\bar{x})^{2}}{N}}=\sqrt{\frac{\Sigma x^{2}}{N}-x^{-2}}$
$\sigma_{y}=\sqrt{\frac{\Sigma(y-\bar{y})^{2}}{N}}=\sqrt{\frac{\Sigma y^{2}}{N}-y^{-2}}$
If $\mathrm{x}-\bar{x}-\bar{y}$ are small fractions, we use
$\mathrm{r}=\frac{\Sigma(x-\bar{x})(y-\bar{y})}{\sqrt{\Sigma(x-\bar{x})^{2}} \sqrt{\Sigma(y-\bar{y})^{2}}}$
If $x, y$ are small numbers, we use
$r=\frac{\Sigma x y-\frac{1}{N} \Sigma x \Sigma y}{\sqrt{\Sigma x^{2}-\frac{1}{N}(\Sigma x)^{2}} \sqrt{\Sigma y^{2}-\frac{1}{N}(\Sigma y)^{2}}}$
If $x, y$ are large numbers, we use assumed mean $A$ and $B$ and $u=x-A, v=y-B$
$\mathrm{r}=\frac{\Sigma u v-\frac{1}{N} \Sigma u \Sigma v}{\sqrt{\Sigma u^{2}-\frac{1}{N}(\Sigma u)^{2}} \sqrt{\Sigma v^{2}-\frac{1}{N}(\Sigma v)^{2}}}$
For example:
Find Karl Pearson's coefficient of correlation between X and Y for the following:

x	5	4	3	2	1
y	4	2	10	8	6

Following problem was given to students on the same concept:
(a) What is the value $\Sigma x y$ in this data?
(b) What is the value $\Sigma x^{2}$?
(c) What is the value of $\Sigma y^{2}$?
OR
What is the value of Karl Pearson's Coefficient of Correlation between x and y?

Answer

Read the text carefully and answer the questions:
In class of Statistics, teacher was discussing the concept of Measures of Correlation, in which he was discussing about Karl Pearson's Coefficient of Correlation. During his class, he discussed the following few points on this:This is the best method for finding correlation between two variables provided the relationship between the two variables is linear. This method is also known as product moment correlation coefficient. Pearson's correlation coefficient may be defined as the ratio of covariance between the two variables to the product of the standard deviations of the two variables.
If the two variables are denoted by x and y and of the corresponding bivariates data are $\left(\mathrm{x}_{\mathrm{i}}, \mathrm{y}_{\mathrm{i}}\right)$ for $\mathrm{i}=1,2,3, \ldots, \mathrm{n}$, then the coefficent of correlation between x and y due to Karl Pearson, is given by:
$r=r_{x y}$
or, $\mathrm{r}_{\mathrm{xy}}=\frac{\operatorname{Cov}(x, y)}{\sqrt{\operatorname{Var} x} \cdot
\sqrt{\operatorname{Var} y}}$
$=\frac{\operatorname{Cov}(x, y)}{\sigma_{x} \cdot \sigma_{y}}$
where,
$\operatorname{cov}(\mathrm{x}, \mathrm{y})=\frac{\Sigma(x-\bar{x})(y-\bar{y})}{N}$
$=\frac{\Sigma x y}{N}-\bar{x} \cdot \bar{y}$
$\sigma_{x}=\sqrt{\frac{\Sigma(x-\bar{x})^{2}}{N}}=\sqrt{\frac{\Sigma x^{2}}{N}-x^{-2}}$
$\sigma_{y}=\sqrt{\frac{\Sigma(y-\bar{y})^{2}}{N}}=\sqrt{\frac{\Sigma y^{2}}{N}-y^{-2}}$
If $\mathrm{x}-\bar{x}-\bar{y}$ are small fractions, we use
$\mathrm{r}=\frac{\Sigma(x-\bar{x})(y-\bar{y})}{\sqrt{\Sigma(x-\bar{x})^{2}} \sqrt{\Sigma(y-\bar{y})^{2}}}$
If $x, y$ are small numbers, we use
$r=\frac{\Sigma x y-\frac{1}{N} \Sigma x \Sigma y}{\sqrt{\Sigma x^{2}-\frac{1}{N}(\Sigma x)^{2}} \sqrt{\Sigma y^{2}-\frac{1}{N}(\Sigma y)^{2}}}$
If $\mathrm{x}, \mathrm{y}$ are large numbers, we use assumed mean A and B and $\mathrm{u}=\mathrm{x}-\mathrm{A}, \mathrm{v}=\mathrm{y}-\mathrm{B}$
$\mathrm{r}=\frac{\Sigma u v-\frac{1}{N} \Sigma u \Sigma v}{\sqrt{\Sigma u^{2}-\frac{1}{N}(\Sigma u)^{2}} \sqrt{\Sigma v^{2}-\frac{1}{N}(\Sigma v)^{2}}}$
For example:
Find Karl Pearson's coefficient of correlation between $X$ and $Y$ for the following:

x	5	4	3	2	1
y	4	2	10	8	6

Following problem was given to students on the same concept:
(i)

x	y	xy
5	4	20
4	2	8
3	10	30
2	8	16
1	6	6
Total		80

(ii)

$x$	$x^2$
5	25
4	16
3	9
2	4
1	1
$\Sigma x^2$	55

(iii)

$y$	$y^2$
4	16
2	4
10	100
8	64
6	36
$\Sigma y^2$	220

$x$	$x^2$	$y$	$y^2$	xy
5	25	4	16	20
4	16	2	4	8
3	9	10	100	30
2	4	8	64	16
1	1	6	36	6
$\sum x$ = 15	$\sum x^2$ = 55	$\sum y$ = 30	$\sum y^2$ = 220	$\sum x y$ = 80

\[\begin{aligned}
& =\frac{\Sigma x y-\frac{1}{N} \Sigma x \Sigma y}{\sqrt{\Sigma x^{2}-\frac{(\Sigma x)^{2}}{N}} \sqrt{\Sigma y^{2}-\frac{(\Sigma y)^{2}}{N}}} \\
& =\frac{80-\frac{1}{5} \times 15 \times 30}{\sqrt{55-\frac{(15)^{2}}{5}}\sqrt{220-\frac{(30)^{2}}{5}}} \\& =\frac{-10}{\sqrt{10} \sqrt{40}}=\frac{-10}{20}=-0.5\end{aligned}\]
Therefore, Karl Pearson's Coefficient of Correlation between x and y is -0.5 .

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Question 44 Marks

Read the text carefully and answer the questions:
Population vs Year graph given below.

(a) In which year the population becomes 110 crores?
(b) Find the equation of line perpendicular to line AB and passing through (1995, 97).
(c) Write the equation of line AB?
OR
Find the slope of line $A B$.

Answer

Read the text carefully and answer the questions:
Population vs Year graph given below.

(i) Equation of line AB is,
$x-2 y=1801$
Putting y $=110$,
$\therefore \mathrm{x}=1801+220$
$\Rightarrow \mathrm{x}=2021$
(ii) $\because$ Slope of $A B=\frac{1}{2}$
Slope of the perpendicular of $\mathrm{AB}=\frac{-1}{\frac{1}{2}}=-2$
$\therefore$ Equation of line perpendicular to AB passing through $(1995,97)$ is
$\Rightarrow \mathrm{y}-97=-2(\mathrm{x}-1995)$
$\Rightarrow \mathrm{y}-97=-2 \mathrm{x}+3990$
$\Rightarrow 2 \mathrm{x}+\mathrm{y}=4087$
(iii)Equation of line $A B$ is,
$\Rightarrow \mathrm{y}-\mathrm{y}_{1}=\mathrm{m}\left(\mathrm{x}-\mathrm{x}_{1}\right)$
$\therefore \mathrm{y}-92=\frac{1}{2}(\mathrm{x}-1985)$
$2 y-184=x-1985$
$\Rightarrow \mathrm{x}-2 \mathrm{y}=1801$
OR
Slope of line $A B$ joining points $A(1985,92)$ and $B(1995,97)$
$\mathrm{m}=\frac{97-92}{1995-1985}=\frac{5}{10}=\frac{1}{2}$

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Applied Maths Case study (4 Marks)

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