Final answer:
The calculated 80% confidence interval for the mean number of apples per tree is (807.73, 842.27), which closely aligns with Option C (808, 842) provided by the student assuming potential rounding differences.
Step-by-step explanation:
To find the 80% confidence interval for the mean number of apples per tree for all trees, given the sample mean (\(\bar{x}\)) is 825 apples, the standard deviation (\(\sigma\)) of the population is 100 apples, and the sample size (\(n\)) is 55 trees, we use the formula for a confidence interval when the population standard deviation is known:
\[CI = \bar{x}\pm z^*\cdot\frac{\sigma}{\sqrt{n}}]\
For an 80% confidence level, the z-score (\(z^*\)) is approximately 1.28. Plugging in the values, we get:
\[CI = 825 \pm (1.28)\cdot\frac{100}{\sqrt{55}}]\
After calculating we find:
\[CI = 825 \pm (1.28)\cdot(13.49)]\
\[CI = 825 \pm 17.27]\
So, the confidence interval is (807.73, 842.27).
This interval does not match exactly any of the provided options A-D, suggesting a potential error in the options provided or a need to check for rounding differences in tables or calculators. Based on calculations, we would expect an interval closest to Option C (808, 842).