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Find the p value following the exponential distribution $mu=3$

Hypothesis testing: find the UMP testPower Function for the uniform distributionCalculate size and power of a given PMFfind distribution of hypothesis testing?Hypothesis testing: normal vs. non-normalp-value, intuition about type-I error=$alpha$Hypothesis Testing: One and Two-Sided TestsBasics of Bayesian hypothesis testingHow to make a Hypothesis test with a Random Variable?p-value of the following test

1

0

$begingroup$

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

$$H_0:muleq 3 quad textvs quad H_1:mu >3$$

The main thing I know is that

$$P(mathrmRe,j mid mu leq 3)=P(Xgeq 3 mid mu leq 3)= e^-1 approx0.36$$

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

hypothesis-testing exponential-distribution p-value

share|cite|improve this question

edited Mar 13 at 18:07

J. W. Tanner

3,4601320

asked Mar 13 at 16:51

Lexie WalkerLexie Walker

1717

$endgroup$

add a comment | 

1

0

$begingroup$

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

$$H_0:muleq 3 quad textvs quad H_1:mu >3$$

The main thing I know is that

$$P(mathrmRe,j mid mu leq 3)=P(Xgeq 3 mid mu leq 3)= e^-1 approx0.36$$

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

hypothesis-testing exponential-distribution p-value

share|cite|improve this question

edited Mar 13 at 18:07

J. W. Tanner

3,4601320

asked Mar 13 at 16:51

Lexie WalkerLexie Walker

1717

$endgroup$

add a comment | 

$begingroup$

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

$$H_0:muleq 3 quad textvs quad H_1:mu >3$$

The main thing I know is that

$$P(mathrmRe,j mid mu leq 3)=P(Xgeq 3 mid mu leq 3)= e^-1 approx0.36$$

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

hypothesis-testing exponential-distribution p-value

share|cite|improve this question

edited Mar 13 at 18:07

J. W. Tanner

3,4601320

asked Mar 13 at 16:51

Lexie WalkerLexie Walker

1717

$endgroup$

share|cite|improve this question

edited Mar 13 at 18:07

J. W. Tanner

3,4601320

asked Mar 13 at 16:51

Lexie WalkerLexie Walker

1717

share|cite|improve this question

edited Mar 13 at 18:07

J. W. Tanner

3,4601320

asked Mar 13 at 16:51

Lexie WalkerLexie Walker

1717

edited Mar 13 at 18:07

J. W. Tanner

3,4601320

edited Mar 13 at 18:07

J. W. Tanner

3,4601320

asked Mar 13 at 16:51

Lexie WalkerLexie Walker

1717

asked Mar 13 at 16:51

Lexie WalkerLexie Walker

1717

1 Answer
1

active

oldest

votes

2

$begingroup$

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:

$$p = intlimits_0^3 e^-mu xd mu = frac1-e^-3xx$$

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

share|cite|improve this answer

edited Mar 13 at 17:52

answered Mar 13 at 17:12

Rohit PandeyRohit Pandey

1,5581023

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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
  

  
  
  
  

add a comment | 

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2

$begingroup$

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:

$$p = intlimits_0^3 e^-mu xd mu = frac1-e^-3xx$$

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

share|cite|improve this answer

edited Mar 13 at 17:52

answered Mar 13 at 17:12

Rohit PandeyRohit Pandey

1,5581023

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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
  

  
  
  
  

add a comment | 

2

$begingroup$

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:

$$p = intlimits_0^3 e^-mu xd mu = frac1-e^-3xx$$

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

share|cite|improve this answer

edited Mar 13 at 17:52

answered Mar 13 at 17:12

Rohit PandeyRohit Pandey

1,5581023

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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
  

  
  
  
  

add a comment | 

$begingroup$

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:

$$p = intlimits_0^3 e^-mu xd mu = frac1-e^-3xx$$

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

share|cite|improve this answer

edited Mar 13 at 17:52

answered Mar 13 at 17:12

Rohit PandeyRohit Pandey

1,5581023

$endgroup$

share|cite|improve this answer

edited Mar 13 at 17:52

answered Mar 13 at 17:12

Rohit PandeyRohit Pandey

1,5581023

answered Mar 13 at 17:12

Rohit PandeyRohit Pandey

1,5581023

answered Mar 13 at 17:12

Rohit PandeyRohit Pandey

1,5581023