Find the p-value (manually) of the followingHypothesis testing. H 0 : μ ≤ 3 vs H 1 : μ &gt; 3The main thing I know is that P ( R e j ∣ μ ≤ 3 ) = P ( X ≥ 3 ∣ μ ≤ 3 ) = e − 1 ≈ 0.36Can I use the z value and use the formula probability of z?

Question

Find the   p-value (manually) of the followingHypothesis testing.      H    0    :  μ  ≤  3    vs        H    1    :  μ  &amp;gt;  3The main thing I know is that  P  (      R    e      j  ∣  μ  ≤  3  )  =  P  (  X  ≥  3  ∣  μ  ≤  3  )  =      e          −      1        ≈  0.36Can I use the   z value and use the formula probability of   z?

Gretchen Bond · Accepted Answer

Your null hypothesis is that your exponential distribution has a rate   μ which is   ≤  3. Your alternate hypothesis is that   μ  ≥  3. Now, you get some observation,   x. What is the probability that this sample is consistent with the null-hypothesis? Meaning, what is the probability that the null hypothesis would generate a sample   ≥  x? Conditional on   μ, this is simply       e          −      μ      x      . Since your null hypothesis is that   μ  ≤  3, you integrate over it to get the   p-value:  p  =      ∫    0    3        e          −      μ      x        d  μ  =            1      −              e                  −          3          x                      x  Now, you can set a threshold on this   p-value and reject the null hypothesis if it is lower than your threshold.

Find the p -value (manually) of the following Hypothesis testing. H 0 </msub> :

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