Determine the critical region for this test at significance level &#x03B1;<!-- α --> = 0.

Pattab 2022-07-01 Answered
Determine the critical region for this test at significance level α = 0.05 .
Let the random variable X be the waiting time until the next student has to go to the toilet. Assume that X has an E x p ( λ ) distribution with unknown λ. We test H 0 : λ = 0.2 against H 1 : λ < 0.2, where we use X as test statistic. Determine the critical region for this test at significance level α = 0.05.
In my opinion I should calculate P ( X < C ) | H 0 ) = 1 e λ x = 1 e 0.2 x So 1 e 0.2 x = 0.05 so x=0.25 and the critical region is ( , 0.25 ], is it the right method to solve this question?
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Answers (1)

Melina Richard
Answered 2022-07-02 Author has 14 answers
In fact, any region R R such that
P λ = 0.2 ( X R ) = 0.05
is a correct answer, in theory. But some of these answers would be absurd in practice.
Since the alternative hypothesis is that λ < 0.2, and since this is equivalent to say that E ( X ) > 1 0.2 = 5, we see that it is only reasonable to reject H 0 when X takes values sensibly greater than 5. That is, the critical region should be
( C , )
for C R such that
P ( X ( C , ) ) = P ( X > C ) = 1 ( 1 e 0.2 C ) = e 0.2 C = 0.05.
So C = 5 ln ( 0.05 ) 14.98 and the critical region is
( 14.98 , + ) ..

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