Chapter 7, Problem 53SE

Aphid infestation of fruit trees can be controlled either by spraying with pesticide or by inundation with ladybugs. In a particular area, four different groves of fruit trees arc selected for experimentation. The first three groves arc sprayed with pesticides 1, 2, and 3, respectively, and the fourth is treated with ladybugs, with the following results on yield:

Treatment	ni = Number of Trees	${\bar{x}}_i$ (Bushels/Tree)	si
1	100	10.5	1.5
2	90	10.0	1.3
3	100	10.1	1.8
4	120	10.7	1.6

Let μ_i= the true average yield (bushels/tree) after receiving the ith treatment. Then

$\theta =\frac{1}{3}({\mu }_1+{\mu }_2+{\mu }_3)-{\mu }_4$

measures the difference in true average yields between treatment with pesticides and treatment with ladybugs. When n₁, n₂, n₃, and n₄ are all large, the estimator $\hat{\theta }$ obtained by replacing each μ_i, by ${\bar{X}}_i$ is approximately normal. Use this to derive a large-sample 100(1 − α)% CI for θ, and compute the 95% interval for the given data.