A new two-stage sampling design for estimating the maximum average time to flower

A horticulturist was considering the number of days (X) each variety took from planting seeds to reach the stage when the first bud appeared for three local marigold varieties. The data X could be recorded with accuracy of one-half day. The primary interest was to estimate the maximum waiting time between "seeding" and "first budding" among three varieties under consideration. It was thought that a 99% confidence interval with width one day would suffice. The horticulturist felt comfortable to assume a normal distribution for the response variable. He provided positive lower bounds for the variances which forced pilot sample sizes to become unequal. We are not aware of any existing methodology with unequal pilot sample sizes that would readily apply here. Accordingly. a new two-stage sampling design was developed and implemented. The gathered data validated all assumptions made during the course of this investigation. Important exact as well as fame-sample properties of the proposed methodology are highlighted (Theorem 1). This methodology is proven to be theoretically superior (Theorem 2) to the existing Methodology for lame sample sizes provided that the pilot sizes could be "chosen" equal. For the experimental data on hand, the superiority of the new methodology has also been indicated (Section 4.3). The solution to the primary estimation problem ultimately led to a natural and yet nontraditional selection problem involving identification of the "Nvorst" marigold variety. For this selection problem, a practical approach is developed (Section 4.4) for evaluating the associated probability of correctly selecting the worst marigold variety.