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 &lt; 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 &lt; 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?

Question

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  &amp;lt;  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  &amp;lt;  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?

Melina Richard · Accepted Answer

In fact, any region   R  ⊂      R   such that      P          λ      =      0.2        (  X  ∈  R  )  =  0.05is a correct answer, in theory. But some of these answers would be absurd in practice.Since the alternative hypothesis is that   λ  &amp;lt;  0.2, and since this is equivalent to say that   E  (  X  )  &amp;gt;      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  &amp;gt;  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  ,  +  ∞  )  ..

Determine the critical region for this test at significance level α = 0.

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