Question

By multiplying each of the numbers 4, 5, 7, 11, 13 by 4 and then adding 7 to each of them, we obtain a new dataset.
Then, the difference between the sample variance of the new dataset and the sample variance of the old dataset is

Answer 1

Answer:

225

Step-by-step explanation:

old variance = 15

New VAriance = old Variance x C^2

So new variance = 16x15=240

Difference is 250-15=225

Answer 2

The difference between the sample variance of the new dataset and the sample variance of the old dataset = 180

Given :

• The numbers 4, 5, 7, 11, 13

• By multiplying each of the numbers 4, 5, 7, 11, 13 by 4 and then adding 7 to each of them, we obtain a new dataset.

To find :

The difference between the sample variance of the new dataset and the sample variance of the old dataset

Solution :

Step 1 of 4 :

Calculate mean of the old dataset

Here the given old data set is 4, 5, 7, 11, 13

Number of observations = n = 5

Sum of the observations

= 4 + 5 + 7 + 11 + 13

= 40

Mean of the data set

$\displaystyle \sf{ = \bar{x} }$

$\displaystyle \sf{ = \frac{Sum \: of \: the \: observations}{Number \: of \: observations} }$

$\displaystyle \sf{ = \frac{40}{5} }$

$\displaystyle \sf{ = 8 }$

Step 2 of 4 :

Calculate the variance of the old dataset

The variance of the old dataset

$\displaystyle \sf{ = \frac{ \sum \: {(x_i - \bar{x} )}^{2} }{n} }$

$\displaystyle \sf{ = \frac{ {(4 - 8)}^{2} +{(5 - 8)}^{2} +{(7 - 8)}^{2} + {(11 - 8)}^{2} +{(13 - 8)}^{2} }{5} }$

$\displaystyle \sf{ = \frac{ {( - 4)}^{2} +{(-3)}^{2} +{( - 1)}^{2} + {(3)}^{2} +{(5)}^{2} }{5} }$

$\displaystyle \sf{ = \frac{16 + 9 + 1 + 9 + 25}{5} }$

$\displaystyle \sf{ = \frac{60}{5} }$

$\displaystyle \sf{ = 12 }$

Step 3 of 4 :

Calculate variance of the new dataset

Let x be random variable representing old data set and y be random variable representing new data set

∴ Var(x) = 12

Since by multiplying each of the numbers 4, 5, 7, 11, 13 by 4 and then adding 7 to each of them, we obtain a new dataset.

∴ y = 4x + 7

$\displaystyle \sf{ \implies }Var(y) = Var(4x + 7)$

$\displaystyle \sf{ \implies }Var(y) = {4}^{2} Var(x)$

$\displaystyle \sf{ \implies }Var(y) = 16 \: Var(x)$

$\displaystyle \sf{ \implies }Var(y) = 16 \times 12$

$\displaystyle \sf{ \implies }Var(y) = 192$

Step 4 of 4 :

Calculate difference between the sample variance of the new dataset and the sample variance of the old dataset

The sample variance of the new dataset = 192

The sample variance of the old dataset = 12

∴ The difference between the sample variance of the new dataset and the sample variance of the old dataset

= 192 - 12

= 180

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