Statistics Test

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Statistics Test - 11

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Weekly Quiz Competition

Question 1
1 / -0

The variance of first $$ 50$$ even natural numbers is

A
$$ \displaystyle \frac{833}{4}$$

B
$$ 833$$

C
$$ 437$$

D
$$ \displaystyle \frac{437}{4}$$

Solution

$${\textbf{Step-1: Use mean formula and find the mean.}}$$
$$\Rightarrow \sigma^2 = \dfrac{1}{n} \sum x_i^2 - \bar{(x)}^2$$

$$\Rightarrow n = 50, \sum x_i = 2 +4+6+8+...+100$$
$${\text{We know that,}}$$
$$\Rightarrow \bar {x} = \dfrac{\sum x_i}{n}$$

$$\Rightarrow \bar {x} = \dfrac{2 +4+6+8+...+100}{50}$$

$$\Rightarrow \bar {x} = \dfrac{50 \times 51}{50}$$ $$[\because \sum 2n = n(n+1)]$$

$$\Rightarrow \bar {x} = 51$$
$${\textbf{Step-2: Put the values in variance formula.}}$$
$$\Rightarrow \sigma^2 = \dfrac{1}{n} \sum x_i^2 - \bar{(x)}^2$$
$$= \dfrac{1}{50} (2^2 +4^2+6^2+8^2+...+100^2) - {(51)}^2$$
$$= \dfrac{1}{50} [({2.1})^2 +({2.2})^2+({2.3})^2+({2.4})^2+...+({2.50})^2] - {(51)}^2$$
$$= \dfrac{1}{50} 2^2[({1})^2 +({2})^2+({3})^2+({4})^2+...+({50})^2] - {(51)}^2 ...(1)$$
$${\text{But,}}$$
$$\Rightarrow ({1})^2 +({2})^2+({3})^2+({4})^2+...+({n})^2 = \dfrac{n(n+1)(2n+1)}{6}$$
$$\Rightarrow ({1})^2 +({2})^2+({3})^2+({4})^2+...+({50})^2 = \dfrac{50(50+1)(2 \times 50+1)}{6}$$
$$\Rightarrow ({1})^2 +({2})^2+({3})^2+({4})^2+...+({50})^2 = \dfrac{50(51)(101)}{6}$$
$${\text{Equation (1) become,}}$$
$$= \dfrac{1}{50} 2^2[\dfrac{50(51)(101)}{6}] - {(51)}^2$$
$$= 34 \times 101 -2601$$
$$ = 3434-2601$$
$$=833$$
$${\textbf{Thus, option B is correct.}}$$
Question 2
1 / -0

The mean of $$x, y, z$$ is $$y$$, then $$x+z =$$

A
$$y$$

B
$$2y$$

C
$$3y$$

D
$$zy$$

Solution

The question tells us that the mean of $$x, y$$ and $$z$$ is $$y$$.

i.e. $$ \dfrac {x+y+z}{3} =y $$

i.e. $$x+z=3y-y$$

$$\rightarrow x+z=2y$$
Question 3
1 / -0

Value of the middle-most observation(s) is called:

A
Mean

B
Median

C
Mode

D
None of these

Solution

To find the Median, place the numbers in value order and find the middle number.
If there are two middle numbers, take the mean of the two numbers and this will be the median of the data set.
The middle most observation of a data series is called the median of the series.
Question 4
1 / -0

The ________ is the difference between the greatest and the least value of the variate.

A
Range

B
Data

C
Average

D
Variance

Solution

$$Range$$ $$as$$ $$the$$ $$name$$ $$indicates$$ $$gives$$ $$us$$ $$all$$ $$the$$ $$area$$ $$available$$ $$under$$ $$light$$ $$and$$ $$hence$$ $$statement$$ $$is$$ $$true.$$
Question 5
1 / -0

The modal value is the value of the variate which divides the total frequency into two equal parts.

A
True

B
False

C
Neither

D
Either

Solution

False. Modal value is the value which occurs maximum number of times in the data.
Question 6
1 / -0

Median divides the total frequency into _____ equal parts.

A
$$2$$

B
$$3$$

C
$$4$$

D
None of these

Solution

The median of the data series is the middle term or the mean of the two middle terms.
Hence, it divides the data series or the frequency of terms into two equal halves.
Question 7
1 / -0

The difference between the maximum and the minimum obervations in data is called the ____________.

A
mean of the data

B
range of the data

C
mode of the data

D
median of the data

Solution

In arithmetic, the range of a set of data is the difference between the largest and smallest values.
So, difference between minimum and maximum values is called range.
Question 8
1 / -0

The variance is the _______ of the standard deviation.

A
Square

B
Cube

C
Square root

D
Cube root

Solution

Standard Deviation:
The Standard Deviation is a measure of how spread out numbers are.
Its symbol is (the greek letter sigma)
The formula is easy: it is the square root of the Variance. So now you ask, "What is the Variance?"
Variance:
The Variance is defined as: The average of the squared differences from the Mean.
Question 9
1 / -0

State true or false:
The mode is the most frequently occurring observation.

A
True

B
False

C
Can't determine

D
None of these

Solution

The observation occurring the most number of times or which has highest frequency is called the mode.
Thus, the given statement is true.
Question 10
1 / -0

Median of $$15, 28, 72, 56, 44, 32, 31, 43\ and\ 51\ is\ 43.$$

A
True

B
False

C
Neither

D
Either

Solution

The terms are: 15, 28, 72, 56, 44, 32, 31, 43 and 51.
Arranging them in ascending order: 15, 28, 31, 32, 43, 44, 51, 56, 72

Since the total number of terms is odd that is 9, therefore the median will be the middle term that is the 5th term which is 43.

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Re-Attempt Weekly Quiz Competition

Statistics Test - 11

Result

Statistics Test - 11

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SHARING IS CARING

Question 1

$$ \displaystyle \frac{833}{4}$$

$$ 833$$

$$ 437$$

$$ \displaystyle \frac{437}{4}$$

Solution

Question 2

$$y$$

$$2y$$

$$3y$$

$$zy$$

Solution

Question 3

Mean

Median

Mode

None of these

Solution

Question 4

Range

Data

Average

Variance

Solution

Question 5

True

False

Neither

Either

Solution

Question 6

$$2$$

$$3$$

$$4$$

None of these

Solution

Question 7

mean of the data

range of the data

mode of the data

median of the data

Solution

Question 8

Square

Cube

Square root

Cube root

Solution

Question 9

True

False

Can't determine

None of these

Solution

Question 10

True

False

Neither

Either

Solution

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