If all the natural numbers between 1 and 20 are multiplied by 3, then what is the variance of the resulting series?

Options

A. 99.75
B. 199.75
C. 299.25
D. 399.25

Explanation

To find the variance of the resulting series when all natural numbers between 1 and 20 are multiplied by 3, we will follow these steps:

### Step 1: Calculate the variance of the natural numbers from 1 to 20.

The formula for variance \( \sigma^2 \) is given by:

\[ \sigma^2 = \frac{\sum x^2}{n} - \left(\frac{\sum x}{n}\right)^2 \]

Where:
- \( n \) is the number of observations.
- \( \sum x \) is the sum of the observations.
- \( \sum x^2 \) is the sum of the squares of the observations.

For natural numbers from 1 to 20:
- \( n = 20 \)

#### Calculate \( \sum x \):
\[ \sum x = 1 + 2 + 3 + ... + 20 = \frac{n(n+1)}{2} = \frac{20(20+1)}{2} = \frac{20 \times 21}{2} = 210 \]

#### Calculate \( \sum x^2 \):
\[ \sum x^2 = 1^2 + 2^2 + 3^2 + ... + 20^2 = \frac{n(n+1)(2n+1)}{6} = \frac{20(20+1)(2 \times 20 + 1)}{6} = \frac{20 \times 21 \times 41}{6} = 2870 \]

### Step 2: Substitute the values into the variance formula.

Now we can substitute \( \sum x \) and \( \sum x^2 \) into the variance formula:

\[ \sigma^2 = \frac{2870}{20} - \left(\frac{210}{20}\right)^2 \]

Calculating each term:

\[ \frac{2870}{20} = 143.5 \]
\[ \left(\frac{210}{20}\right)^2 = 10.5^2 = 110.25 \]

Now, substituting these values back into the variance formula:

\[ \sigma^2 = 143.5 - 110.25 = 33.25 \]

### Step 3: Calculate the new variance after multiplying by 3.

When each observation is multiplied by a constant \( k \), the variance of the new series is given by:

\[ \sigma'^2 = k^2 \cdot \sigma^2 \]

In this case, \( k = 3 \):

\[ \sigma'^2 = 3^2 \cdot 33.25 = 9 \cdot 33.25 = 299.25 \]

### Final Answer:
The variance of the resulting series after multiplying all natural numbers between 1 and 20 by 3 is **299.25**.

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