Measures of Dispersion Test

Result

Measures of Dispersion Test - 9

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Weekly Quiz Competition

Question 1
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Find the Standard Deviation of the following data:
$$5,9,8,12,6,10,6,8$$

A
$$2.14$$

B
$$2.16$$

C
$$2.15$$

D
$$2.17$$

Solution

Here $$n=8$$.
Mean, $$u=\dfrac{\sum x_i}{n}=\dfrac{64}{8}=8$$
Value of Deviations $$(x_i-u)$$ are
$$-3,1,0,4,-2,2,-2,0$$
Variance, $$\sigma ^2=\dfrac{\sum(x_i-u)^2}{n}\\=\dfrac{(-3-8)^2-(1-8)^2-(0-8)^2-(4-8)^2-(-2-8)^2-(2-8)^2-(-2-8)^2-(0-8)^2}{8}\\=\dfrac{38}{8}=4.75$$.
Standard Deviation, $$S.D=\sqrt{\sigma ^2}$$
$$\therefore S.D=\sqrt{4.75}=2.17$$
Ans- Option $$D$$.
Question 2
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Find the variance of first $$10$$ multiples of $$3$$.

A
$$72.65$$

B
$$74.05$$

C
$$74.25$$

D
$$73.85$$

Solution

First $$10$$ multiples of $$3$$ are $$3,6,9...30$$.
This is an A.P.
$$sum=\dfrac n2 (a+l)= \dfrac {10}{2} \times (3+30)$$
$$\therefore sum=165$$.
Mean, $$u=\dfrac {sum}{n}=\dfrac{165}{10}$$.
Variance, $$\sigma ^2=\dfrac{\sum(x_i ^2)}{n}-u^2$$.
$$\therefore \sigma ^2=\dfrac{3^2+6^2+...30^2}{10}-{16.5}^2$$.
$$\therefore \sigma ^2=\dfrac{3\times(1^2+2^2+...10^2)}{10}-{16.5}^2$$.
$$\therefore \sigma ^2=\dfrac{9\times 10\times (10+1)\times (2\times 10+1)}{6\times 10}-{16.5}^2$$.
$$\therefore \sigma ^2=346.5-272.25$$
$$\therefore \sigma ^2=74.25$$
Ans-Option $$C$$.
Question 3
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Standard Deviation of first n natural numbers.

A
$$\sqrt{\dfrac{n^2+1}{6}}$$

B
$$\sqrt{\dfrac{n^2-1}{12}}$$

C
$$\sqrt{\dfrac{n^2+1}{12}}$$

D
$$\sqrt{\dfrac{n^2-1}{6}}$$

Solution

Mean, $$u=\dfrac{(1+2+3...+n)}{n}$$

$$\therefore u=\dfrac 12 (n+1)$$

Variance, $$\sigma ^2=\dfrac{\sum(x_i-u)^2}{n}=\dfrac{\sum x_i^2}{n}-u^2$$

$$\therefore \sigma ^2= \dfrac{\sum n^2}{n}-{\dfrac 12 (n+1)}^2$$

$$\therefore \sigma ^2= \dfrac 1n \dfrac{n(n+1)(2n+1)}{n}-({\dfrac 12 (n+1)})^2$$

$$\therefore \sigma ^2=\dfrac{n^2-1}{12}$$
Standard Deviation, $$S.D=\sqrt{\sigma ^2}$$
$$\therefore S.D=$$\sqrt{\dfrac{n^2-1}{12}}$$$$

Ans- Option $$B$$.
Question 4
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Variance of first $$n$$ natural numbers.

A
$$\dfrac{n^2-1}{6}$$

B
$$\dfrac{n^2+1}{6}$$

C
$$\dfrac{n^2-1}{12}$$

D
$$\dfrac{n^2+1}{12}$$

Solution

Mean, $$u=\dfrac{(1+2+3...+n)}{n}$$

$$\therefore u=\dfrac 12 (n+1)$$

Variance, $$\sigma ^2=\dfrac{\sum(x_i-u)^2}{n}=\dfrac{\sum x_i^2}{n}-u^2$$

$$\therefore \sigma ^2= \dfrac{\sum n^2}{n}-{\dfrac 12 (n+1)}^2$$

$$\therefore \sigma ^2= \dfrac 1n \dfrac{n(n+1)(2n+1)}{n}-({\dfrac 12 (n+1)})^2$$

$$\therefore \sigma ^2=\dfrac{n^2-1}{12}$$
Ans-Option $$C$$
Question 5
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Standard Error of Mean is defined as ________.

A
Standard deviation of the sampling distribution of mean

B
Average of sampling distribution of mean

C
Inter-Quartile range of sampling distribution of mean

D
Correlation co-efficient between the sampling distribution of mean and mean
Question 6
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Directions For Questions

Factory $$A$$ $$B$$
Number of Workers $$4000$$ $$5000$$
Mean Wages $$3500$$ $$3500$$
Variance in wages $$64$$ $$81$$

...view full instructions

Which factory has more variation in wages?

A
$$A$$

B
$$B$$

C
Equal Variation

D
Cannot be determined.

Solution

$$(\sigma _A) ^2=64$$ and $$(\sigma _B) ^2=81$$
$$\because S.D=\sqrt {\sigma ^2}$$
$$\therefore S.D_A=8$$ and $$S.D_B=9$$
Since monthly mean wages in two factories are same, and $$ S.D_A < S.D_B$$.
Thus, the factory $$B$$ has more variation in wages.
Ans-Option $$B$$.
Question 7
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The Standard error is denoted by__________.

A
$$S_{yx} =$$ S.E. of estimate of y for given x

B
$$S_{xy} =$$ S.E. of estimate of x for given y

C
both (A) and (B)

D
none of the above

Solution

Standard error is a statistical measure that shows the accuracy by which a sample represents the population. Standard error is denoted by both the equations mentioned in option a and b.
Question 8
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A sampling distribution of means is______________.

A
appropriate

B
a probability distribution that indicates the likely values the various sample means take on all of the possible samples

C
calculated as calculating mean of the population

D
all of the above

Solution

A sampling distribution is the probability distribution of a given statistics based on random samples, it shows the likely values of the various sample means.
Question 9
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The variance of six observations is given as $$16$$ and mean is $$8$$. If each observation is multiplied by $$3$$.
Find the new variance:

A
$$155$$

B
$$124$$

C
$$177$$

D
$$144$$

Solution

Mean, $$u=8$$.
So, $$\dfrac{x_1+x_2...x_6}{6}=8$$.
$$\therefore x_1+x_2...x_6=48$$.
Multiplying the above equation by $$3$$ on both the sides we get,
$$3x_1+3x_2...3x_6=48\times 3$$
$$\therefore u'=\dfrac{3x_1+3x_2....3x_6}{6}=\dfrac{48 \times 3}{6}$$
$$\therefore u'=24$$
Thus, new mean, $$u'=24$$.
$$\therefore$$ New variance, $$\sigma ^2=\dfrac{(3x_1)^2+...(3x_6)^2}{6}-(24)^2$$
$$\therefore \sigma ^2=9\times \dfrac{(x_1)^2+....(x_6)^2}{6}-576$$
$$\therefore \sigma ^2=9\times \dfrac{480}{16}-576$$
$$\therefore \sigma ^2=720-576$$
$$\therefore \sigma ^2=144$$
Thus, New Variance is $$144$$.
Ans-Option $$D$$
Question 10
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Which of the following is true with respect to range?

A
Higher value of Range implies higher dispersion and vice-versa

B
Range is unduly affected by extreme values.

C
It is not based on all the values.

D
All of these

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Re-Attempt Weekly Quiz Competition

Factory	$$A$$	$$B$$
Number of Workers	$$4000$$	$$5000$$
Mean Wages	$$3500$$	$$3500$$
Variance in wages	$$64$$	$$81$$

Measures of Dispersion Test - 9

Result

Measures of Dispersion Test - 9

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Question 1

$$2.14$$

$$2.16$$

$$2.15$$

$$2.17$$

Solution

Question 2

$$72.65$$

$$74.05$$

$$74.25$$

$$73.85$$

Solution

Question 3

$$\sqrt{\dfrac{n^2+1}{6}}$$

$$\sqrt{\dfrac{n^2-1}{12}}$$

$$\sqrt{\dfrac{n^2+1}{12}}$$

$$\sqrt{\dfrac{n^2-1}{6}}$$

Solution

Question 4

$$\dfrac{n^2-1}{6}$$

$$\dfrac{n^2+1}{6}$$

$$\dfrac{n^2-1}{12}$$

$$\dfrac{n^2+1}{12}$$

Solution

Question 5

Standard deviation of the sampling distribution of mean

Average of sampling distribution of mean

Inter-Quartile range of sampling distribution of mean

Correlation co-efficient between the sampling distribution of mean and mean

Question 6

$$A$$

$$B$$

Equal Variation

Cannot be determined.

Solution

Question 7

$$S_{yx} =$$ S.E. of estimate of y for given x

$$S_{xy} =$$ S.E. of estimate of x for given y

both (A) and (B)

none of the above

Solution

Question 8

appropriate

a probability distribution that indicates the likely values the various sample means take on all of the possible samples

calculated as calculating mean of the population

all of the above

Solution

Question 9

$$155$$

$$124$$

$$177$$

$$144$$

Solution

Question 10

Higher value of Range implies higher dispersion and vice-versa

Range is unduly affected by extreme values.

It is not based on all the values.

All of these

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