Question 9 A study of peach trees found that the average number of peaches per tree was 525 . The standard deviation of the population is 140 peaches per tree. A scientist wishes to find the \( 90 \% \) confidence interval for the mean number of peaches per tree. How many trees does she need to sample to obtain an average accurate to within 18 peaches per tree? 100 128 61 165

To determine the required sample size for a confidence interval, we use the formula \( n = \left( \frac{Z \times \sigma}{E} \right)^2 \), where \( Z \) is the z-score corresponding to the desired confidence level, \( \sigma \) is the standard deviation, and \( E \) is the margin of error. For a 90% confidence level, the z-score is approximately 1.645. Plugging in the values: \[ n = \left( \frac{1.645 \times 140}{18} \right)^2 \approx 128 \] So, the scientist needs to sample **128 trees** to ensure her estimate of the average number of peaches per tree is accurate within 18 peaches. To ensure accuracy in calculations like this, always double-check your usage of the z-score for the confidence level and the correct interpretation of the margin of error. A common mistake is forgetting to square the entire fraction when determining \( n \) or miscalculating the z-score. Also, consider if your desired margin of error is appropriate; smaller margins often require much larger sample sizes!