Understanding the Arithmetic Mean Problem

This question asks us to find the new arithmetic mean of a set of observations after applying a specific change: multiplying every observation by 5. We are given the original mean and the number of observations.

Given Information

• Number of observations (\(N\)): 200
• Original arithmetic mean (\(\bar{X}\)): 60
• Operation applied to each observation: Multiplication by 5

Arithmetic Mean Basics

The arithmetic mean, often called the average, is calculated by summing up all the values in a dataset and then dividing by the count of those values. The formula is:

\(\bar{X} = \frac{\sum_{i=1}^{N} x_i}{N}\)

Where '\(\sum_{i=1}^{N} x_i\)' represents the sum of all observations (\(x_1, x_2, \dots, x_N\)) and \(N\) is the total number of observations.

Step-by-Step Calculation

First, let's use the given information to find the sum of the original observations.

We know that \(\bar{X} = 60\) and \(N = 200\). Plugging these into the formula:

\(60 = \frac{\sum_{i=1}^{200} x_i}{200}\)

To find the sum of the original observations, we multiply the mean by the number of observations:

\(\text{Sum of original observations} = \sum_{i=1}^{200} x_i = 60 \times 200 = 12000\)

Next, consider the effect of multiplying each observation by 5. Let the new observations be \(x'_i\). Then, \(x'_i = 5 \times x_i\) for each \(i\) from 1 to 200.

The sum of the new observations is:

\(\text{Sum of new observations} = \sum_{i=1}^{200} x'_i = \sum_{i=1}^{200} (5 x_i)\)

Using the properties of summation, we can pull the constant factor 5 out:

\(\sum_{i=1}^{200} (5 x_i) = 5 \times \sum_{i=1}^{200} x_i\)

We already calculated the sum of the original observations (\(12000\)), so:

\(\text{Sum of new observations} = 5 \times 12000 = 60000\)

Finally, to find the new arithmetic mean (\(\bar{X}_{new}\)), we divide the sum of the new observations by the number of observations (which is still 200):

\(\bar{X}_{new} = \frac{\text{Sum of new observations}}{N} = \frac{60000}{200}\)

\(\bar{X}_{new} = 300\)

Using Properties of Mean for a Quicker Solution

There's a useful property of the arithmetic mean that simplifies this type of problem significantly. The property states:

If each observation in a dataset is multiplied by a constant value \(k\), the mean of the dataset is also multiplied by the same constant \(k\).

Mathematically, if \(x'_i = k \times x_i\), then the new mean \(\bar{X}_{new}\) is:

\(\bar{X}_{new} = k \times \bar{X}\)

Applying this property to our problem:

• Original Mean (\(\bar{X}\)) = 60
• Constant Multiplier (\(k\)) = 5

New Mean (\(\bar{X}_{new}\)):

\(\bar{X}_{new} = 5 \times 60\)

\(\bar{X}_{new} = 300\)

Both methods confirm that the new arithmetic mean is 300.