text: # 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!
explanation: The document contains a detailed math question related to hypothesis testing and the Poisson distribution. The question is about determining if a filtration system is working correctly based on the number of particles found in water samples. The document also includes an answer and further explanation to clarify the solution.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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!
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a detailed math question related to hypothesis testing and the Poisson distribution. The question is about determining if a filtration system is working correctly based on the number of particles found in water samples. The document also includes an answer and further explanation to clarify the solution.
  
  Label: question-and-answer