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 sample size needed to achieve a specified margin of error for a confidence interval, we can use the formula for the sample size \( n \): \[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \] where: - \( Z \) is the Z-score corresponding to the desired confidence level, - \( \sigma \) is the population standard deviation, - \( E \) is the margin of error. For a \( 90\% \) confidence level, the Z-score is approximately \( 1.645 \). Given: - \( \sigma = 140 \) peaches, - \( E = 18 \) peaches. Now, we can plug in the values: \[ n = \left( \frac{1.645 \cdot 140}{18} \right)^2 \] Calculating the numerator: \[ 1.645 \cdot 140 = 230.3 \] Now, divide by the margin of error: \[ \frac{230.3}{18} \approx 12.7389 \] Now, square this value: \[ n \approx (12.7389)^2 \approx 162.2 \] Since the sample size must be a whole number, we round up to the next whole number: \[ n = 163 \] However, since this value is not among the options provided, we should check the calculations again. Let's recalculate: 1. Calculate \( \frac{1.645 \cdot 140}{18} \): \[ 1.645 \cdot 140 = 230.3 \] \[ \frac{230.3}{18} \approx 12.7389 \] 2. Now square it: \[ (12.7389)^2 \approx 162.2 \] Since we need to round up, we get \( n = 163 \). None of the provided options (100, 128, 61, 165) match exactly, but the closest option is **165**. Thus, the answer is: **165**