Suppose you are planning a greenhouse experiment on growth of pepper plants. You will grow n individually potted seedlings in standard soil and another n seedlings in specially treated soil. After 21 days, you will measure Y = total stem length (cm) for each plant. If the effect of the soil treatment is to increase the population mean stem length by 2 cm, you would like to have a 90% chance of rejecting H0 with a one-tailed t test. Data from a pilot study (e.g., the data in Exercise 2.S.2) on 15 plants grown in standard soil give  = 12.5 cm and s = 0.8 cm.
(a) Suppose you plan to test at α = 0.05. Use the pilot information to determine what value of n you should use.
(b) What conditions are necessary for the validity of the calculation in part (a)? Which of these can be checked (roughly) from the data of the pilot study?
(c) Suppose you decide to adopt a more conservative posture and test at α = 0.01. What value of n should you use?