A pilot study indicated that the weight of a species is found to have a standard deviation of 20 g . The size of a sample that would be needed to estimate the population mean to within 2.5 g with \( 95 \% \) confidence is: \( \begin{array}{llll}\text { A. } 190 & \text { B. } 239 & \text { C. } 246 & \text { D. } 256\end{array} \)

To determine the required sample size, you can use the formula for sample size estimation based on the desired confidence level, population standard deviation, and margin of error. For a 95% confidence interval, the z-score is approximately 1.96. The formula to calculate the sample size \( n \) is: \[ n = \left( \frac{z \cdot \sigma}{E} \right)^2 \] Where: - \( z \) is the z-score (1.96 for 95% confidence) - \( \sigma \) is the population standard deviation (20 g) - \( E \) is the margin of error (2.5 g) Plugging in the numbers: \[ n = \left( \frac{1.96 \cdot 20}{2.5} \right)^2 \] Calculating that gives: \[ n = \left( \frac{39.2}{2.5} \right)^2 = (15.68)^2 \approx 246.1 \] Thus, rounding up, you would need a sample size of at least 247. Therefore, the answer that best fits from the options is **C. 246** when considering standard practice of rounding down to the nearest whole number for practical purposes. When it comes to statistics, did you know that the concept of sampling dates back to ancient times? Even the Roman empire utilized basic sampling methods to assess crop yields and census data. It’s fascinating how the groundwork for modern statistical techniques has such historical roots! If you’re diving deeper into sample size calculations, don't forget the importance of understanding power analysis. Knowing how many participants are needed not only saves resources but also ensures your study has a robust chance of detecting real effects. Books on experimental design can really enhance your grasp of this essential topic!