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Hypothesis Testing $X sim Exp(textmean =frac1theta)$: Rejection Region and Power Function

0

$begingroup$

I am working on a problem and I would like to have some advice. The following is the given information.

1), $X$ is exponential with mean $frac1theta$.

2), $H_0: theta =5$ vs $H_1: theta<5$

3), $alpha=.05$

and our goal is

a), Find the rejection region.

b), Find the power function and determine where it is maximized.

What I have tried so far is this.

Edited:

$$
Pr[X > k | theta = 5] = alpha$$
so

$$e^-k(5) = alpha$$
and
$$beginalign
k & = -frac15ln (alpha) \
& sim .599
endalign$$

So I am thinking that the rejection region is when the sample is greater than or equal to $.599 space$ .

Moving on,

$$beginalign
gamma(theta) & = Pr[X > .599|theta < 5] \
& = e^-.599 space theta
endalign$$

I am more used to using concrete numbers rather than theory, so I am not too confident if I am doing this right.

I would really appreciate any help.

statistics hypothesis-testing

share|cite|improve this question

edited Apr 8 at 20:45

hyg17

asked Apr 8 at 7:31

hyg17hyg17

2,00422044

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The function $gamma(theta)=1-e^-0.0103theta$ is monotonically increasing in $theta$. And $gamma(0)=0$, in particular. So the closer $theta$ to 5, the larger power function is. It is meaningless and indicates that the rejection region is selected incorrectly: the great difference of hypothesis should provide large value of pover function, and not vice versa. Note that expected value of $X$ is reciprocal to $theta$. So the smaller $theta$, the greater should be $X$. So good rejection region should be $X>k$. For this case $sup gamma(theta) = gamma(0)$ although $theta>0$.
  $endgroup$
  – NCh
  Apr 8 at 17:52
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thank you for your comment. Does it look better now?
  $endgroup$
  – hyg17
  Apr 8 at 20:46
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, I like it better
  $endgroup$
  – NCh
  Apr 9 at 1:46
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Perhaps see this Q&A or look at Wikipedia on exponential distribution.
  $endgroup$
  – BruceET
  Apr 9 at 14:42
  

  
  
  
  

add a comment | 

0

$begingroup$

I am working on a problem and I would like to have some advice. The following is the given information.

1), $X$ is exponential with mean $frac1theta$.

2), $H_0: theta =5$ vs $H_1: theta<5$

3), $alpha=.05$

and our goal is

a), Find the rejection region.

b), Find the power function and determine where it is maximized.

What I have tried so far is this.

Edited:

$$
Pr[X > k | theta = 5] = alpha$$
so

$$e^-k(5) = alpha$$
and
$$beginalign
k & = -frac15ln (alpha) \
& sim .599
endalign$$

So I am thinking that the rejection region is when the sample is greater than or equal to $.599 space$ .

Moving on,

$$beginalign
gamma(theta) & = Pr[X > .599|theta < 5] \
& = e^-.599 space theta
endalign$$

I am more used to using concrete numbers rather than theory, so I am not too confident if I am doing this right.

I would really appreciate any help.

statistics hypothesis-testing

share|cite|improve this question

edited Apr 8 at 20:45

hyg17

asked Apr 8 at 7:31

hyg17hyg17

2,00422044

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The function $gamma(theta)=1-e^-0.0103theta$ is monotonically increasing in $theta$. And $gamma(0)=0$, in particular. So the closer $theta$ to 5, the larger power function is. It is meaningless and indicates that the rejection region is selected incorrectly: the great difference of hypothesis should provide large value of pover function, and not vice versa. Note that expected value of $X$ is reciprocal to $theta$. So the smaller $theta$, the greater should be $X$. So good rejection region should be $X>k$. For this case $sup gamma(theta) = gamma(0)$ although $theta>0$.
  $endgroup$
  – NCh
  Apr 8 at 17:52
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thank you for your comment. Does it look better now?
  $endgroup$
  – hyg17
  Apr 8 at 20:46
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, I like it better
  $endgroup$
  – NCh
  Apr 9 at 1:46
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Perhaps see this Q&A or look at Wikipedia on exponential distribution.
  $endgroup$
  – BruceET
  Apr 9 at 14:42
  

  
  
  
  

add a comment | 

$begingroup$

I am working on a problem and I would like to have some advice. The following is the given information.

1), $X$ is exponential with mean $frac1theta$.

2), $H_0: theta =5$ vs $H_1: theta<5$

3), $alpha=.05$

and our goal is

a), Find the rejection region.

b), Find the power function and determine where it is maximized.

What I have tried so far is this.

Edited:

$$
Pr[X > k | theta = 5] = alpha$$
so

$$e^-k(5) = alpha$$
and
$$beginalign
k & = -frac15ln (alpha) \
& sim .599
endalign$$

So I am thinking that the rejection region is when the sample is greater than or equal to $.599 space$ .

Moving on,

$$beginalign
gamma(theta) & = Pr[X > .599|theta < 5] \
& = e^-.599 space theta
endalign$$

I am more used to using concrete numbers rather than theory, so I am not too confident if I am doing this right.

I would really appreciate any help.

statistics hypothesis-testing

share|cite|improve this question

edited Apr 8 at 20:45

hyg17

asked Apr 8 at 7:31

hyg17hyg17

2,00422044

$endgroup$

share|cite|improve this question

edited Apr 8 at 20:45

hyg17

asked Apr 8 at 7:31

hyg17hyg17

2,00422044

share|cite|improve this question

edited Apr 8 at 20:45

hyg17

asked Apr 8 at 7:31

hyg17hyg17

2,00422044

asked Apr 8 at 7:31

hyg17hyg17

2,00422044

asked Apr 8 at 7:31

hyg17hyg17

2,00422044