Statistics Test

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Statistics Test - 32

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Weekly Quiz Competition

Question 1
1 / -0

The mean of a distribution is $$14$$ and standard deviation is $$5$$. What is the value of the coefficient of variation?

A
$$57.7\%$$

B
$$45.7\%$$

C
$$35.7\%$$

D
None of these

Solution

Coefficient of variation is given by $$CV = \dfrac{SD}{Mean}\times 100 $$
$$\Rightarrow \dfrac{5}{14}\times 100 = 35.7\%$$
Question 2
1 / -0

Find variance for following data:
Class $$0-10$$ $$10-20$$ $$20-30$$ $$30-40$$ $$40-50$$ $$50-60$$ $$60-70$$
frequency $$6$$ $$5$$ $$8$$ $$15$$ $$7$$ $$6$$ $$3$$

A
$$14.75$$

B
$$15.75$$

C
$$17.75$$

D
$$16.75$$

Solution

$$\bar { x } =\cfrac { \sum { Fx } }{ \sum { F } } =\cfrac{1670}{50}=33.4$$
$${\sigma}^{2}=\cfrac { \sum { F{ ( }x-\bar { x } )^{ 2 } } }{ \sum { F } } =\cfrac{14025.6}{50}=280.51$$
$$\sigma=16.75$$
Question 3
1 / -0

The degree to which numerical data tend to spread about value is called

A
mean

B
variation

C
median

D
mode

Solution

Variance gives the degree to which the numerical data tend to spread about the value.

Variance is given by $$\dfrac{\sum x^2}{n}-(\dfrac{\sum x}{n})^2$$
Question 4
1 / -0

The SD of following:
Age $$20-25$$ $$25-30$$ $$30-35$$ $$35-40$$ $$40-45$$ $$45-50$$
Number of persons $$170$$ $$110$$ $$80$$ $$45$$ $$40$$ $$35$$

A
$$62.62$$

B
$$56.56$$

C
$$7.93$$

D
$$9.24$$

Solution

$$\bar { x } =\cfrac { \sum { Fx } }{ \sum { F } }=\cfrac{14500}{480}=30.2 $$.
$${\sigma}^{2}=\cfrac { \sum { F{ ( }x-\bar { x } )^{ 2 } } }{ \sum { F } }=\cfrac{40983.25}{480}=85.38 $$
$$\sigma=9.24$$
Question 5
1 / -0

If $$x $$ is increased by $$k$$ then $$\sigma$$ changes to

A
$$k+\sigma$$

B
$$k\sigma$$

C
$$k\sqrt{\sigma}$$

D
remains unchanged

Solution

By the properties of standard deviation,
$$\sigma(x+k)=\sigma(x)$$
Therefore, by the addition of the constant, the standard deviation remains unchanged.
Question 6
1 / -0

Find variance for following data:
Marks $$0-4$$ $$5-9$$ $$10-14$$ $$15-19$$ $$20-24$$ $$25-29$$ $$30-34$$ $$35-39$$
Frquency $$2$$ $$5$$ $$7$$ $$13$$ $$21$$ $$16$$ $$8$$ $$3$$

A
$$7$$

B
$$55$$

C
$$64$$

D
$$78$$

Solution

$$\bar { x } =\cfrac { \sum { Fx } }{ \sum { F } } =\cfrac{1605}{75}=21.4$$
$${\sigma}^{2}=\cfrac { \sum { F{ ( }x-\bar { x } )^{ 2 } } }{ \sum { F } }=\cfrac{4798}{75}=63.97\approx 64$$
Question 7
1 / -0

The variance and SD of the following is
$$x$$ $$4.5$$ $$14.5$$ $$24.5$$ $$34.5$$ $$44.5$$ $$54.5$$ $$64.5$$
$$f$$ $$1$$ $$5$$ $$12$$ $$22$$ $$17$$ $$9$$ $$4$$

A
$$176,13$$

B
$$175.9,13.26$$

C
$$8.56,13$$

D
$$4.1,12.13$$

Solution

$$\bar { x } =\cfrac { \sum { Fx } }{ \sum { F } } =\cfrac{2635}{70}=37.64$$
$${\sigma}^{2}=\cfrac { \sum { F{ ( }x-\bar { x } )^{ 2 } } }{ \sum { F } } =\cfrac{12308.6}{70}=175.9$$
$$\therefore \sigma=13.26$$ .
Question 8
1 / -0

If $$x $$ is multiplied by $$k$$ then $$\sigma$$ changes to

A
$$k^2\sigma$$

B
$$k\sigma$$

C
$$k\sqrt{\sigma}$$

D
$$k(\sigma)^2$$

Solution

By the properties of standard deviation,
$$\sigma(kx)=|k|\sigma(x)$$
Therefore, by multiplying the $$x$$ by $$k$$, SD changes to $$k\sigma$$
Question 9
1 / -0

Standard Deviation of n observations $$a_1, a_2, a_3 .....a_n$$ is $$\sigma$$ Then the standard deviation of the observations $$\lambda a_1, \lambda a_2, ..., \lambda a_n$$ is

A
$$\lambda \sigma$$

B
$$-\lambda \sigma$$

C
$$|\lambda| \sigma$$

D
$${\lambda}^n \sigma$$

Solution

Let the mean of original data is $$\mu$$
Then mean of observation if all the terms are multiplied by the constant $$\lambda$$ will be $$\lambda \mu$$
If $$\sigma$$ be the standard deviation of original observation , then
$$\sigma=\sqrt{\dfrac{1}{n}\displaystyle \sum_{i=1}^n (x_i-\mu)^2 }$$
Now let the standard deviation of new observation is $$\sigma'$$, then
$$\sigma'=\sqrt{\dfrac{1}{n}\displaystyle \sum_{i=1}^n (\lambda x_i-\lambda\mu)^2 }$$
$$=\sqrt{\dfrac{1}{n}\displaystyle \sum_{i=1}^n \lambda^2( x_i-\mu)^2 }$$
$$=|\lambda|\sqrt{\dfrac{1}{n}\displaystyle \sum_{i=1}^n( x_i-\mu)^2 }=|\lambda |\sigma$$
Note that: standard deviation can never be negative
Question 10
1 / -0

The mean deviation from the data $$3, 10, 10, 4, 7, 10, 5$$.

A
$$3$$

B
$$2$$

C
$$3.75$$

D
$$2.57$$

E
None of these

Solution

Given data is, $$3,10,10,4,7,10,5$$.
Mean, $$\mu=\dfrac{\displaystyle \sum x_i}{N}=\dfrac{3+10+10+4+7+10+5}{7}=\dfrac{49}{7}=7$$
Thus mean deviation $$=\dfrac{\displaystyle \sum|x_i-\mu|}{N}=\dfrac{|3-7|+3(10-7)+|4-7|+(7-7)+|5-7|}{7}$$ $$=\dfrac{4+9+3+0+2}{7}=2.57$$

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Re-Attempt Weekly Quiz Competition

Class	$$0-10$$	$$10-20$$	$$20-30$$	$$30-40$$	$$40-50$$	$$50-60$$	$$60-70$$
frequency	$$6$$	$$5$$	$$8$$	$$15$$	$$7$$	$$6$$	$$3$$

Age	$$20-25$$	$$25-30$$	$$30-35$$	$$35-40$$	$$40-45$$	$$45-50$$
Number of persons	$$170$$	$$110$$	$$80$$	$$45$$	$$40$$	$$35$$

Marks	$$0-4$$	$$5-9$$	$$10-14$$	$$15-19$$	$$20-24$$	$$25-29$$	$$30-34$$	$$35-39$$
Frquency	$$2$$	$$5$$	$$7$$	$$13$$	$$21$$	$$16$$	$$8$$	$$3$$

$$x$$	$$4.5$$	$$14.5$$	$$24.5$$	$$34.5$$	$$44.5$$	$$54.5$$	$$64.5$$
$$f$$	$$1$$	$$5$$	$$12$$	$$22$$	$$17$$	$$9$$	$$4$$

Statistics Test - 32

Result

Statistics Test - 32

Score

-

Rank

-

SHARING IS CARING

Question 1

$$57.7\%$$

$$45.7\%$$

$$35.7\%$$

None of these

Solution

Question 2

$$14.75$$

$$15.75$$

$$17.75$$

$$16.75$$

Solution

Question 3

mean

variation

median

mode

Solution

Question 4

$$62.62$$

$$56.56$$

$$7.93$$

$$9.24$$

Solution

Question 5

$$k+\sigma$$

$$k\sigma$$

$$k\sqrt{\sigma}$$

remains unchanged

Solution

Question 6

$$7$$

$$55$$

$$64$$

$$78$$

Solution

Question 7

$$176,13$$

$$175.9,13.26$$

$$8.56,13$$

$$4.1,12.13$$

Solution

Question 8

$$k^2\sigma$$

$$k\sigma$$

$$k\sqrt{\sigma}$$

$$k(\sigma)^2$$

Solution

Question 9

$$\lambda \sigma$$

$$-\lambda \sigma$$

$$|\lambda| \sigma$$

$${\lambda}^n \sigma$$

Solution

Question 10

$$3$$

$$2$$

$$3.75$$

$$2.57$$

None of these

Solution

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