To determine the necessary sample size for reducing the maximum error $E$ of an estimate for the mean $\mu$ to one-fifth its original size, the sample size $n$ must increase by a factor of 25. This is due to the relationship where $E$ is inversely proportional to the square root of $n$. Given the formula $E=z_{\alpha/2}\cdot\frac{\sigma}{\sqrt{n}}$, if the original sample size is 200, the new required sample size would be 5000. Thus, the calculation confirms that to achieve this reduction in error, the sample size must indeed be 5000. Understanding these principles is crucial for accurately estimating confidence intervals.
