Statistics Test

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Statistics Test - 21

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Weekly Quiz Competition

Question 1
1 / -0

If the SD of a set of observation is $$8$$ and each observation is divided by $$-2$$, then the SD of the new set of observations will be.

A
$$-4$$

B
$$-8$$

C
$$8$$

D
$$4$$

Solution

We know that, if X and Y are two variables such that $$\displaystyle Y=\frac{X}{a}, a\neq 0$$
Then, $$\sigma _{Y}=\frac{1}{|a|}\sigma_{X}$$, where $$\sigma_{Y}$$ is the SD of new observations, $$\sigma_{X}$$ is the SD for old observations and $$a$$ is the factor from which each observation is being divided by
$$\therefore$$ SD of the new observations $$=\displaystyle\frac{8}{|-2|}=4$$.
Question 2
1 / -0

Range and standard deviation are similar in that each looks

A
The difference between the high and the low scores.

B
The numerical value that occurs the most often.

C
How spread out the data is.

D
The central score.

Solution

Range:- In mathematical terms, range is the distance between the highest value & lowest value
The range also represents the variability of the data. Datasets with a large range are said to have large variability. While data sets with smaller range are said to have small variability. Generally, smaller variability is better because it represents more precise measurements & yields more accounts analysis
Standard deviation:- While the range is the good gange of the variability of data, there is more accurante & useful one:
standard deviation:
S.D. is a reliable method for determing how variable the data is for both a sample & a population. The deviation is too much a score varies from the to all mean of the data.
Question 3
1 / -0

The mean deviation of the numbers $$3,4,5,6,7$$ from mean is

A
$$25$$

B
$$5$$

C
$$1.2$$

D
$$0$$

Solution

$$\overset { \_ }{ X } =\dfrac { 3+4+5+6+7 }{ 5 } =5$$
So mean deviation from mean is
$$=\dfrac { |3-5|+|4-5|+|5-5|+|6-5|+|7-5| }{ 5 } \\ =\dfrac { 2+1+0+1+2 }{ 5 } =1.2$$
So option $$C$$ is correct.
Question 4
1 / -0

Mean of $$9$$ observations was founded to be $$35$$. Later on, it was detected that an observation $$81$$ was misread as $$18$$, then the correct mean of the observations is ____________.

A
$$40$$

B
$$41$$

C
$$42$$

D
$$43$$

Solution

Mean of $$9$$ observations $$= 35$$
Sum of $$9$$ observations $$= 35\times9 = 315$$

Since $$81$$ was misread as $$18$$
New sum $$= 315-18+81= 378$$

The correct mean $$= \dfrac{378}{9} = 42$$
Question 5
1 / -0

Mean deviation of $$6, 8, 12, 15, 10, 9$$ through mean is:

A
$$10$$

B
$$2.33$$

C
$$2$$

D
None of these

Solution

$$\overline { x } =\cfrac { 6+8+12+15+10+9 }{ 6 } =\cfrac { 60 }{ 6 } $$
$$ \overline { x } = 10$$
Mean deviation about mean $$= \cfrac { 1 }{ 6 } \left[ \left| 6-10 \right| +\left| 8-10 \right| +\left| 12-10 \right| +\left| 15-10 \right| +\left| 10-10 \right| +\left| 9-10 \right| \right]$$
$$ = \cfrac { 1 }{ 6 } (14)$$
$$ = 2.33$$
Question 6
1 / -0

The sum of the deviations of the variates from the arithmetic mean is always.

A
$$+1$$

B
$$0$$

C
$$-1$$

D
Real number

Solution

The sum of the deviations of a given set of observations from their arithmetic mean is always zero. It is due to the property that the arithmetic mean is characterised as the centre of gravity. i.e. sum of positive deviation from the mean is equal to the sum of negative deviations.
For example:
$$3,4,6,8,14$$
$$\overline{x}=\dfrac{3+4+6+8+14}{5}=7$$

$$x_{i}$$ $$x_{i}-\overline{x}$$
$$3$$ $$-4$$
$$4$$ $$-3$$
$$6$$ $$-1$$
$$8$$ $$1$$
$$14$$ $$7$$

$$\sum (x_{i} - \overline{x})=-8+8=0$$
Hence, the sum of the deviations about mean is $$0$$.
Question 7
1 / -0

Size 2 3 4 5 6 7 8
Frequency 10 12 25 20 25 15 11
The mode of the following distribution is ______.

A
$$2$$

B
$$8$$

C
Both $$4$$ and $$6$$

D
$$5$$

Solution

Mode is that observation which have highest frequency. Since, both $$4$$ and $$6$$ have highest frequency i.e. $$25$$ and $$25$$, they are the mode of the given distribution.
Hence, option (C) is correct.
Question 8
1 / -0

If the median of the data $$6,7,x-2,x,18,21$$ written in ascending order is $$16$$, then the variance of the data is

A
$$30\dfrac{1}{5}$$

B
$$31\dfrac{1}{3}$$

C
$$32\dfrac{1}{2}$$

D
$$33\dfrac{1}{3}$$

Solution

The given observations are,
$$6,7,x-2,x,18,21$$
Number of observations $$(n)=6$$ [ even ]
$$\Rightarrow$$ $$Median=\dfrac{3^{rd}obs+4^{th}obs}{2}$$

$$\Rightarrow$$ $$16=\dfrac{x-2+x}{2}$$

$$\Rightarrow$$ $$32=x-2+x$$
$$\Rightarrow$$ $$32=2x-2$$
$$\Rightarrow$$ $$34=2x$$
$$\therefore$$ $$x=17$$
$$\therefore$$ The observations becomes,
$$6,7,15,17,18,21$$
$$\overline{x}=\dfrac{6+7+15+17+18+21}{6}=\dfrac{84}{6}=14$$

$$Variance(\sigma^2)=\dfrac{(6-14)^2+(7-14)^2+(15-14)^2+(17-14)^2+(18-14)^2+(21-14)^2}{6}$$

$$=\dfrac{(-8)^2+(-7)^2+(1)^2+(3)^2+(4)^2+(7)^2}{6}$$

$$=\dfrac{64+49+1+9+16+49}{6}$$

$$=\dfrac{188}{6}$$

$$=\dfrac{94}{3}$$

$$=31\dfrac{1}{3}$$
Question 9
1 / -0

The mean salary paid per week to $$1000$$ employees of an establishment was found to be Rs. $$900$$. Later on, it was discovered that the salaries of two employees were wrongly recorded as Rs. $$750$$ and Rs. $$365$$ instead of Rs. $$570$$ and Rs. $$635$$. Find the corrected mean salary.

A
$$900.90$$

B
$$1,115$$

C
$$1,225$$

D
$$900.09$$

Solution

We first find the corrected sum of observation.
Corrected sum of observations$$=$$(Sum of total incorrect observation)$$-$$(Sum of incorrect data)$$+$$(Sum of correct data).
$$=900\times 1,000-(750+365)+(570+635)=9,00,090$$
$$\therefore$$ Corrected Mean $$=Rs.\dfrac{9,00,090}{1,000}=Rs.900.09$$
Question 10
1 / -0

Find the arithmetic mean of numbers from $$1$$ to $$9$$.

A
$$9$$

B
$$5$$

C
$$8$$

D
$$3$$

Solution

The numbers from $$1$$ to $$9$$ are $$1,2,3,4,5,6,7,8,9$$
$$ A.M. =\dfrac{\text{Sum of observations}}{\text{Number of observations}}= \dfrac {1+2+3+4+5+6+7+8+9}{9} = \dfrac {45}{9} = 5 $$

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Statistics Test - 21

Result

Statistics Test - 21

Score

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SHARING IS CARING

Question 1

$$-4$$

$$-8$$

$$8$$

$$4$$

Solution

Question 2

The difference between the high and the low scores.

The numerical value that occurs the most often.

How spread out the data is.

The central score.

Solution

Question 3

$$25$$

$$5$$

$$1.2$$

$$0$$

Solution

Question 4

$$40$$

$$41$$

$$42$$

$$43$$

Solution

Question 5

$$10$$

$$2.33$$

$$2$$

None of these

Solution

Question 6

$$+1$$

$$0$$

$$-1$$

Real number

Solution

Question 7

$$2$$

$$8$$

Both $$4$$ and $$6$$

$$5$$

Solution

Question 8

$$30\dfrac{1}{5}$$

$$31\dfrac{1}{3}$$

$$32\dfrac{1}{2}$$

$$33\dfrac{1}{3}$$

Solution

Question 9

$$900.90$$

$$1,115$$

$$1,225$$

$$900.09$$

Solution

Question 10

$$9$$

$$5$$

$$8$$

$$3$$

Solution

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