The lifetime in hours of each bulb manufactured by a particular company follows an independent exponential distribution with mean \theta. To test the hypothesis H_0 : \theta = 2000 versus H_1 : \theta = 1000, an experiment sets up an experiment with 50 bulbs with 5 bulbs in each of 10 different locations to examine their lifetimes.
To get quick preliminary results, the experimenter decides to stop the experiment as soon as one bulb fails at each location. Let Y_i denote the lifetime of the first bulb to fail at location i, i = 1, 2, ..., 10. Obtain the most powerful test of size 0.05 to test H_0 versus H_1 based on the available lifetimes. If the available data is
510, 9752, 5650, 12385, 230, 4225, 860, 300, 3000 and 1500,
give your conclusion.
[Given that : P(\chi_{(20)}^{2} < 10.851) = 0.05]