**Core Concept**
When conducting statistical analysis, the minimum sample size is a critical factor in ensuring the reliability of the results. The minimum sample size is determined by the desired level of precision, which is often represented by the range of allowable error (also known as the margin of error). In general, a smaller range of allowable error requires a larger sample size to achieve the desired level of precision.

**Why the Correct Answer is Right**
Doubling the range of allowable error increases the acceptable margin of error, allowing for less precise estimates. This reduction in precision requirements means that a smaller sample size can still provide a reliable estimate of the population parameter. Mathematically, this is represented by the formula for sample size: n = (Z^2 * σ^2) / E^2, where n is the sample size, Z is the Z-score corresponding to the desired confidence level, σ is the population standard deviation, and E is the range of allowable error. When E is doubled, the required sample size decreases by a factor of 4, assuming all other variables remain constant.

**Why Each Wrong Option is Incorrect**
**Option A:** Doubling the range of allowable error would actually require a larger sample size, not a smaller one. This is because a larger margin of error would increase the uncertainty of the estimate, requiring more data points to achieve the desired level of precision.

**Option B:** This option is incorrect because the relationship between sample size and range of allowable error is not linear. Doubling the range of allowable error does not simply double the sample size; it requires a quadrupling of the sample size to maintain the same level of precision.

**Option C:** This option is incorrect because it suggests that the sample size remains the same when the range of allowable error is doubled. However, as explained above, a larger margin of error requires a larger sample size to maintain the same level of precision.

**Option D:** This option is incorrect because it suggests that the sample size is reduced by a factor of 2, rather than a factor of 4, when the range of allowable error is doubled.

**Clinical Pearl / High-Yield Fact**
When designing a study or experiment, it's essential to carefully consider the trade-off between sample size and precision. Increasing the range of allowable error can reduce the required sample size, but may also compromise the reliability of the results.

**Correct Answer: C. The required sample size decreases by a factor of 4.