A researcles wants to determie the \( 99 \% \) cenfidence internal ertee mean number of heurs spent doing community serviee. How large a sample she The resmaneler select so that the estimate will be within 1.5 heurs of the popakition mean? Assumet the standard deviation for heurs spent per wee adults doins canmunity seivice is 3 . \[ \begin{array}{l} E=1.5 \\ 2=2.571 \end{array} \]

To determine the sample size required for a \( 99\% \) confidence interval that is within \( 1.5 \) hours of the population mean, you can use the formula for sample size: \[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \] Where: - \( Z \) is the z-score for a \( 99\% \) confidence level, which is approximately \( 2.576 \). - \( \sigma \) is the population standard deviation (in this case, \( 3 \)). - \( E \) is the margin of error (in this case, \( 1.5 \)). Substituting these values: \[ n = \left( \frac{2.576 \cdot 3}{1.5} \right)^2 \] This will yield the necessary sample size. After calculating, you'll find the result should be rounded up to the nearest whole number, since you can’t sample a fraction of a person. Additionally, when conducting research, having a robust sample size is crucial to enhance the validity of your findings. It boosts your confidence that the sample mean accurately reflects the population mean. A larger sample size leads to smaller confidence intervals, which means more precise estimates and less variability in your results. It's a win-win for researchers! Remember to also account for practical considerations—like time and resources—when determining your sample size, as balancing rigor and feasibility will lead to far more successful research outcomes.