Question

QUESTION 1

1. A company that manufactures electronic components wants to test, using a level of significance of 0.1, whether decreasing the environmental temperature decreases the time to failure of one of the components they manufacture. A random sample of 15 components was taken; 9 were tested at 80° F (X₁), and 6 were tested at 120° F (X₂). The results are given below.

   State the null and alternative hypotheses.

Answer 1

$\text{Failure times usually follow exponential distributions. Let the }80^{\circ}F\\ \text{observations be: }X_{1},\dots,X_{9}\stackrel{iid}{\sim}Exp(1/\theta_{1})\\ \text{The }120^{\circ}\text{ observations: }Y_{1},\dots,Y_{6}\stackrel{iid}{\sim}Exp(1/\theta_{2})\\ \text{We can then test: }\\ H_{0}:\theta_{1}=\theta_{2}\text{ Vs. }H_{1}:\theta_{1}<\theta_{2}\\ \\$