# Thread: hypothesis testing, poisson distribution

1. ## hypothesis testing, poisson distribution

At one stage of a water treatment process the number of particles of foreign matter per litre present in the water has a poisson distribution with mean 10. The water then enters a filtration bed which should extract 75% of foreign matter. The manager of the treatment works orders a study into the effectiveness of this filtration bed. 20 samples, each of 1 litre, are taken from the water and 64 particles of foreign matter are found. Using a suitable approximation test, at the 5% level of significance, weather or not there is evidence that the filter bed is failing to work properly.
I know what approximation to use, ( the normal distribution) however I can't seem to work what value the of $\displaystyle \lambda$ is in 20 samples.

Any help appreciated,
Thank you.

H0: The filtration system is operating correctly
H1: The filtration system is not operating correctly

Therefore under H0, which we will assume for our model,
the matter present in a litre of water should be Poisson distributed with lambda = 2.5

Since we can add the means of collected Poisson distributions,
for the twenty litres collectively, we should have a Poisson distribution with lambda = 2.5 + 2.5 + 2.5 ...
= 20*2.5
= 50

3. Originally Posted by mdruett
H0: The filtration system is operating correctly
H1: The filtration system is not operating correctly

Therefore under H0, which we will assume for our model,
the matter present in a litre of water should be Poisson distributed with lambda = 2.5

Since we can add the means of collected Poisson distributions,
for the twenty litres collectively, we should have a Poisson distribution with lambda = 2.5 + 2.5 + 2.5 ...
= 20*2.5
= 50
thank you, but I don't understand where you got 2.5 from? If the null hypothesis is true, than there is a mean of 10 foreign particles per litre, and we have 20 samples, so I thought you multiply.

can you please explain further? how did you get 2.5?

thank you

4. ## Still short of a decent title

H0: is that the filtration system is working properly.

There are a mean of 10 particles per litre before it enters the filtration system.

The filtration system removes 75% of this, under H0

Therefore after the filtration system (where you'd have to take the samples from if you were going to check whether it was working)
there are only 25% of the particles left under H0

ie the mean should be = 0.25 * 10
This is lamdba

5. Originally Posted by mdruett
H0: is that the filtration system is working properly.

There are a mean of 10 particles per litre before it enters the filtration system.

The filtration system removes 75% of this, under H0

Therefore after the filtration system (where you'd have to take the samples from if you were going to check whether it was working)
there are only 25% of the particles left under H0

ie the mean should be = 0.25 * 10
This is lamdba
I understand now, was not thinking of the 75% taken out,

thanks!
The text passage contains a detailed discussion of a hypothesis testing problem involving a Poisson distribution and the effectiveness of a filtration system. It includes a question about the value of the parameter $\lambda$ for 20 samples and an explanation of how to calculate it.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Thread: hypothesis testing, poisson distribution

1. ## hypothesis testing, poisson distribution

At one stage of a water treatment process the number of particles of foreign matter per litre present in the water has a poisson distribution with mean 10. The water then enters a filtration bed which should extract 75% of foreign matter. The manager of the treatment works orders a study into the effectiveness of this filtration bed. 20 samples, each of 1 litre, are taken from the water and 64 particles of foreign matter are found. Using a suitable approximation test, at the 5% level of significance, weather or not there is evidence that the filter bed is failing to work properly.
I know what approximation to use, ( the normal distribution) however I can't seem to work what value the of $\displaystyle \lambda$ is in 20 samples.

Any help appreciated,
Thank you.

H0: The filtration system is operating correctly
H1: The filtration system is not operating correctly

Therefore under H0, which we will assume for our model,
the matter present in a litre of water should be Poisson distributed with lambda = 2.5

Since we can add the means of collected Poisson distributions,
for the twenty litres collectively, we should have a Poisson distribution with lambda = 2.5 + 2.5 + 2.5 ...
= 20*2.5
= 50

3. Originally Posted by mdruett
H0: The filtration system is operating correctly
H1: The filtration system is not operating correctly

Therefore under H0, which we will assume for our model,
the matter present in a litre of water should be Poisson distributed with lambda = 2.5

Since we can add the means of collected Poisson distributions,
for the twenty litres collectively, we should have a Poisson distribution with lambda = 2.5 + 2.5 + 2.5 ...
= 20*2.5
= 50
thank you, but I don't understand where you got 2.5 from? If the null hypothesis is true, than there is a mean of 10 foreign particles per litre, and we have 20 samples, so I thought you multiply.

can you please explain further? how did you get 2.5?

thank you

4. ## Still short of a decent title

H0: is that the filtration system is working properly.

There are a mean of 10 particles per litre before it enters the filtration system.

The filtration system removes 75% of this, under H0

Therefore after the filtration system (where you'd have to take the samples from if you were going to check whether it was working)
there are only 25% of the particles left under H0

ie the mean should be = 0.25 * 10
This is lamdba

5. Originally Posted by mdruett
H0: is that the filtration system is working properly.

There are a mean of 10 particles per litre before it enters the filtration system.

The filtration system removes 75% of this, under H0

Therefore after the filtration system (where you'd have to take the samples from if you were going to check whether it was working)
there are only 25% of the particles left under H0

ie the mean should be = 0.25 * 10
This is lamdba
I understand now, was not thinking of the 75% taken out,

thanks!
user
The text passage contains a detailed discussion of a hypothesis testing problem involving a Poisson distribution and the effectiveness of a filtration system. It includes a question about the value of the parameter $\lambda$ for 20 samples and an explanation of how to calculate it.

TYPE: SINGLE_QUESTION_WITH_ANSWER