\begin{tabular}{l} 4.6.2 Teet (CsT): statistical Analysis \\ Question 3 of 25 \\ If a sample mean is 94 , which of the following is most likely the range of \\ possible values that best describes an estimate for the population mean? \\ A. \( (82,114) \) \\ B. \( (78,110) \) \\ C. \( (80,112) \) \\ D. \( (76,108) \) \\ \hline\end{tabular}

To estimate the population mean based on a sample mean, we typically use a confidence interval. A confidence interval provides a range of values within which we expect the true population mean to lie, with a certain level of confidence (commonly 95%). **Given:** - **Sample Mean (\(\bar{x}\))**: 94 **Options:** - A. (82, 114) - B. (78, 110) - C. (80, 112) - D. (76, 108) **Approach:** 1. **Symmetry Around the Sample Mean:** - A well-constructed confidence interval is symmetric around the sample mean. This means the distance from the sample mean to the lower bound should be approximately equal to the distance from the sample mean to the upper bound. 2. **Calculate the Center of Each Interval:** - **Option A:** \((82 + 114)/2 = 98\) - **Option B:** \((78 + 110)/2 = 94\) - **Option C:** \((80 + 112)/2 = 96\) - **Option D:** \((76 + 108)/2 = 92\) Only **Option B** has a center equal to the sample mean (94), ensuring symmetry. 3. **Range Consistency:** - All options span a total range of 32 units. However, symmetry is crucial for an accurate confidence interval. **Conclusion:** **Option B** provides a symmetric interval around the sample mean of 94, making it the most appropriate estimate for the population mean. **Answer:** **B. \( (78, 110) \)**