8.15 Many bulk products, such as iron ore, coal, and raw sugar, are sampled for quality by a method that requires many small samples to be taken periodically as the material is moving along a conveyor belt. The small samples are then aggregated and mixed to form one composite sample. Let Yi denote the volume of the i th small sample from a particular lot, and suppose Y1, …,Yn constitutes a random sample with each Yi having mean and variance . The average volume of the samples can be set by adjusting the size of the sampling device.

Suppose the variance of sampling volumes is known to be approximately 4 for a particular situation (measurements are in cubic inches). It is required that the total volume of the composite sample exceed 200 cubic inches with probability approximately 0.95 when n  50 small samples are selected. Find a setting for that will allow the sampling requirements to be satisfied.