Minimizing Margin of Error

Solution & Explanation

To solve 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. The margin of error (ME) in a statistical sample is often calculated using the formula:

$\text{Margin of Error} = Z \times \frac{\sigma}{\sqrt{n}}$

Where:

• $Z$ is the Z-score, which corresponds to the desired confidence level.
• $\sigma$ is the population standard deviation.
• $n$ is the sample size.

Given:

• Initial margin of error $ME_1 = 3$
• Desired margin of error $ME_2 = 0.3$

Objective:

• Find the additional number of samples, $k$, required to achieve the desired margin of error.

Steps to Solution:

1. Initial Equation Setup:

   • From the initial condition:
     $3 = Z \times \frac{\sigma}{\sqrt{n}}$

2. Desired Condition Setup:

   • For the desired margin of error:
     $0.3 = Z \times \frac{\sigma}{\sqrt{n+k}}$

3. Express $n$ in terms of $Z$ and $\sigma$:

   • Rearranging the initial equation, we have:
     $n = \left( \frac{Z \times \sigma}{3} \right)^2$

4. Express $n+k$ in terms of $Z$ and $\sigma$:

   • Rearranging the desired equation, we have:
     $n+k = \left( \frac{Z \times \sigma}{0.3} \right)^2$

5. Find the relationship between $n$ and $n+k$:

   • From the two equations:
     $\left( \frac{Z \times \sigma}{0.3} \right)^2 = \frac{1}{0.09} \times \left( \frac{Z \times \sigma}{3} \right)^2$
   • Simplifying, we find:
     $n+k = \frac{1}{0.09} \times n = 100n$

6. Calculate $k$:

   • Substitute $n+k = 100n$ into the equation:
     $k = 100n - n = 99n$

Therefore, to reduce the margin of error from 3 to 0.3, the sample size needs to be increased by a factor of 100, meaning that 99 additional $n$ samples are required.

Conclusion

The additional number of samples required to achieve a margin of error of 0.3, given an initial margin of error of 3, is 99 times the original sample size. This significant increase in sample size reflects the inverse square root relationship between sample size and margin of error.