To cut the margin of error in half while maintaining a fixed confidence level and population standard deviation, what should the sample size be?

To determine how to cut the margin of error in half while maintaining a fixed confidence level and population standard deviation, it's critical to understand the relationship between sample size and the margin of error in statistical inference.

The margin of error (E) in estimating a population parameter using a sample can be expressed in relation to the sample size (n) as follows: [ E = \frac{z \cdot \sigma}{\sqrt{n}} ] where ( z ) is the z-score corresponding to the desired confidence level, ( \sigma ) is the population standard deviation, and ( n ) is the sample size.

To cut the margin of error in half, you would set up the equation: [ \frac{E}{2} = \frac{z \cdot \sigma}{\sqrt{n'}} ] where ( n' ) is the new sample size required to achieve this new margin of error.

When you compare the original margin of error to the new one, the relationship can be manipulated to show: [ \frac{E}{2} = \frac{z \cdot \sigma}{\sqrt{n'}} ] If you substitute the original margin of error ( E ) into the