A very thorough hobby farmer knows that he will plant 492 soy plant seeds in 2022. Based on his past experience, 91% of all soy plant seeds he has planted have sprouted.

a) What is the theoretical probability that at least 450 of the seeds sprout?

b) Normal approximation:
i. What conditions are required to use this approximation?
ii. Are those conditions satisfied?
iii. Compute the approximate probability regardless of whether the conditions are met.

c) The farmer randomly samples 100 seeds and then records whether each seed sprouted or not
in a spreadsheet (see soy.csv: in the file, there are 492 records, of which 30 are marked with 0, 462 are marked with 1). Did a larger proportion of seeds sprout in 2022 compared to past years?

d) Provide an 85% non-parametric confidence interval using 1,000 bootstrapped samples for the true proportion of seeds that sprouted in 2022.