Reducing the Margin of Error

Solution & Explanation

To tackle the problem of reducing the margin of error from 3 to 0.3, we need to understand the relationship between the margin of error and the sample size.

Understanding Margin of Error:

The margin of error (MoE) is a measure of the amount of random sampling error in a survey's results. It can be mathematically expressed as:

$\text{MoE} = z \times \frac{\sigma}{\sqrt{n}}$

• z is the z-score, which corresponds to the desired confidence level.
• σ is the standard deviation of the population.
• n is the sample size.

From the formula, we see that the margin of error is inversely proportional to the square root of the sample size $n$. This means that as the sample size increases, the margin of error decreases.

Problem Breakdown:

Given:

• Initial margin of error (MoE1) = 3
• Target margin of error (MoE2) = 0.3

We need to find the new sample size $n_2$ that will achieve the target margin of error.

Mathematical Derivation:

The relationship between the margin of error and the sample size can be expressed as:

$\frac{\text{MoE1}}{\text{MoE2}} = \frac{\sqrt{n_2}}{\sqrt{n_1}}$

Plugging in the given values:

$\frac{3}{0.3} = \frac{\sqrt{n_2}}{\sqrt{n_1}}$

Simplifying:

$10 = \frac{\sqrt{n_2}}{\sqrt{n_1}}$

Squaring both sides:

$100 = \frac{n_2}{n_1}$

This implies:

$n_2 = 100 \times n_1$

Conclusion:

To reduce the margin of error from 3 to 0.3, the sample size must be increased by a factor of 100. If the original sample size is $n_1$, the new sample size $n_2$ required is $100 \times n_1$. Therefore, you would need 99 times the original sample size in addition to the initial sample size to achieve the desired margin of error.