Tag: statistics

Questions Related to statistics

Calculate the coefficient of correlation between $x$ and $y$ for the data

x 1 2 3 4 5 6 7 8 9 10
y 3 10 5 1 2 9 4 8 7 6
statistics linear correlation coefficient of correlation correlation coefficient correlation coefficients

  1. $0.12$

  2. $0.19$

  3. $0.22$

  4. $0.62$

Show answer
Correct Option: B
Explanation:
$x\\ 1\\ 2\\ 3\\ 4\\ 5\\ 6\\ 7\\ 8\\ 9\\ 10\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \sum { x=55 } $         $y\\ 3\\ 10\\ 5\\ 1\\ 2\\ 9\\ 4\\ 8\\ 7\\ 6\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \sum { y=55 } $                 $X=x-\overline { x } \\ -4.5\\ -3.5\\ -2.5\\ -1.5\\ -0.5\\ \quad 0.5\\ \quad 1.5\\ \quad 2.5\\ \quad 3.5\\ \quad 4.5\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \quad 0$                          $XY\\ \quad 11.25\\ -15.75\\ \quad 1.25\\ \quad 6.75\\ \quad 1.75\\ \quad 1.75\\ -1.25\\ \quad 6.25\\ \quad 1.75\\ \quad 2.25\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \quad 16\\ $                   ${ \quad X }^{ 2 }\\ 20.25\\ 12.25\\ 06.25\\ 02.25\\ 00.25\\ 00.25\\ 02.25\\ 06.25\\ 12.25\\ 20.25\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ 82.50$

${ \quad Y }^{ 2 }\\ 06.25\\ 20.25\\ 00.25\\ 12.25\\ 12.25\\ 02.25\\ 06.25\\ 02.25\\ 00.25\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \quad 82.50$

Therefore, $\overline { x } =\cfrac { 55 }{ 10 } \\ \quad =5.5$
$\cfrac { \sum { y }  }{ 10 } =5.5$

Therefore, $r=\cfrac { \sum { XY }  }{ \sqrt { \sum { { X }^{ 2 }\sum { { Y }^{ 2 } }  }  }  } \\ =\cfrac { 16 }{ 82.5 } \\ =0.19$

The greatest possible number of points of intersection of 8 straight lines and $4$ circles is $104$.

statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

According to question,

There are 8 lines, for two lines meet in a point,

=>  $^8C _2\times 1= \dfrac{8.7}{1.2}=28$

Line and circle meet in two points,

=>$(^8C _1\times ^4C _1) \times 2 =64$

Two circles meet in two points,

=>  $(^4C _2)\times2 =\dfrac{4.3}{1.2}.2= 104$

Solve:$\dfrac{2}{2}+\dfrac{3}{3}+\dfrac{4}{4}+$...... + upto $1000$ terms= ?

statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

  1. $1000$

  2. twice of $500$

  3. four times of $250$

  4. All of these

Show answer
Correct Option: D
Explanation:

All the terms simplify to $1$ which is being added $1000$ times.
Therefore, the final answer is $1000$ and it can be seen that all the options are correct.

There are 6 equally spaced points A, B, C, D, E and F marked on a circle with radius R. How many convex pentagons of distinctly different areas can be drawn using these points advertises?

statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

  1. $^6P _5$

  2. $1$

  3. $55$

  4. $42$

Show answer
Correct Option: B

From 0 to 9 , four digited numbers can be formed such that
the digits  are in ascending order is

statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

  1. ${}^{10}{P _4}$

  2. ${}^{10}{C _4}$

  3. ${}^{10}{P _4} - {}^9{P _3}$

  4. ${}^{10}{C _4} - {}^9{C _3}$

Show answer
Correct Option: B
Explanation:

Selection of$ 4$ digits out of$ 10$ (including 0)=$ ^{10}C _4$


A point $(a, b)$ is called a good point if both $a$ and $b$ are integers. Number of good points on the curve $xy$ $=$ $225$ are

statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

  1. 20

  2. 18

  3. 16

  4. 14

Show answer
Correct Option: B
Explanation:

The order pair $(x, y)$ satisfying $xy=225$ are $(1, 225), (3, 75) (5, 45), (9, 25), (15, 15)$. Order can be changed in the first four pairs and both $x$ and $y$ can be negative also, so the no. of pairs $=2(2\times 4+1)=18$

A graph may be defined as a set of points connected by lines called edges. Every edge connects a pair of points. Thus, a triangle is a graph with 3 edges and 3 points. The degree of a point is the number of edges connected to it. For example, a triangle is agraph with three points of degree 2 each. Consider a graph with 12 points. It is possible to reach any point from any other point through a sequence of edges. The number of edges "e" in the graph must satisfy the condition

statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

  1. $11 \leq e \leq 66$

  2. $10 \leq e \leq 66$

  3. $11 \leq e \leq 65$

  4. $0 \leq e \leq 11$

Show answer
Correct Option: A
Explanation:

(A) Since every edge connects a pair of points, the given 12 points have to be joined using lines. We may have minimum number of edges if all the 12 points are collinear.
No. of edges in this particular case 
$=12-1=11$
Maximum number of edges are possible when all the 12 points are non-collinear. In this particular case number of different straight lines that can be formed using 12 points which is equal to $^12C _{2}$
$=\frac{12\times 11}{2}=66$
Therefore, following inequality holds for "e"
$11 \leq e  \leq 66$

Total number of ways of selecting two numbers from the set ${1,2,3,...90}$ so that their sum is divisible by $3$ is

statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

  1. $885$

  2. $1335$

  3. $1770$

  4. $3670$

Show answer
Correct Option: B

If the letter of word $MOTHER$ are written in all possible orders and these words are written out as in a dictionary. Find the rank of the word $MOTHER$

statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

  1. $307$

  2. $308$

  3. $309$

  4. $120$

Show answer
Correct Option: A

There are $6$ boxes numbered $1, 2 ....... 6$. Each box is to be filled up either with a red or a green ball in such a way that at least $1$ box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is:

statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

  1. $5$

  2. $21$

  3. $33$

  4. $60$

Show answer
Correct Option: B
Explanation:

(B) The number of ways in which 1 green ball can be put $=6$ . The number of ways in which two green balls can be put such that the boxes are consecutive 
$=5$ $(i.e., (1, 2),(2, 3),(3, 4),(4, 5),(5, 6))$

Similarly, the number of ways in which three green balls can be put 
$=4( i.e. (1, 2, 3),(2, 3, 4),(3, 4, 5),(4, 5, 6))$
$\cdots \cdots \cdots \cdots \cdots $ and so on.
$\therefore $ Total number of ways of doing this
$=6+5+4+3+2+1=21$