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I want to find the $p$-value (manually) of the following Hypothesis testing.

$$H_0:\mu\leq 3 \quad \text{vs} \quad H_1:\mu >3$$

The main thing I know is that

$$P(\mathrm{Re}\,j \mid \mu \leq 3)=P(X\geq 3 \mid \mu \leq 3)= e^{-1} \approx0.36$$

Can I use the $z$ value and use the formula probability of $z$? Or from where can I start?

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Your null hypothesis is that your exponential distribution has a rate $\mu$ which is $\leq 3$. Your alternate hypothesis is that $\mu \geq 3$. 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 $\geq x$? Conditional on $\mu$, this is simply $e^{-\mu x}$. Since your null hypothesis is that $\mu \leq 3$, you integrate over it to get the p-value:

$$p = \int\limits_0^3 e^{-\mu x}d \mu = \frac{1-e^{-3x}}{x}$$

Now, you can set a threshold on this p-value and reject the null hypothesis if it is lower than your threshold.

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  • $\begingroup$ Okay, if i did get this right: I need to find an observation x such that my p-value is small? Because I know that my p-value for this test has to be close to zero. $\endgroup$ – Lexie Walker Mar 13 at 17:25
  • $\begingroup$ xAlso, it shouldn't be divided by $x$ instead of 3? $\endgroup$ – Lexie Walker Mar 13 at 17:30
  • $\begingroup$ Yes, sorry.. fixed the typo. Yes, you need to have a very large $x$ for your p-value to be small. The larger the $x$, the smaller the chance an exponential with rate $<3$ generated it. $\endgroup$ – Rohit Pandey Mar 13 at 17:53

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