S2 question. Exercise 7D question 12.

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Student7654
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So, I did the question and I came to a similar answer to the book and the same conclusion however our approaches were different and the answers were not completely parallel, and this is due to the angle we took the question from.

Heres the question: 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. Twenty samples, each 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, whether or not there is evidence that the filter bed is failing to work properly.

In the book, they base the normal distribution on a derived poisson. They say the average amount of particles passing through the filtration bed is 50 and get this by finding the average amount of particles in 20 litres (20*10) then multiplying it by 0.25 (the proportion that is said to get through). They then approximate using a normal distribution defined by X-N(50,50) and then get an answer of 0.0282.Now what I did, was find the average of particles in 20 litres (200), and then model the entire thing as a binomial distribution using the 0.75 as a probability of a particle being removed. Defined X-B(200,0.75).I then approximated using a normal distribution and that's where my method and the books method diverged in terms of values. My derived normal distribution was X-N(50,37.5). As you can see, the variances differ. Therefore, this brought me to a slightly smaller value of Z(-2.20) and my final probability was 0.0139 (I flipped it around and said 136 particles did not get filtered then found P(X<136.5).

Both of our resultant probabilities were smaller than the specified 0.05 significance level therefore we both arrived at the same conclusion, that the filtration bed was not working as well as he thought.

Would I get penalised for taking a difference approach about the question and not getting the same answer? As far as I'm concerned my method is as legit as theirs.What do you guys think?

Thanks in advance.
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Ayman!
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(Original post by Student7654)
So, I did the question and I came to a similar answer to the book and the same conclusion however our approaches were different and the answers were not completely parallel, and this is due to the angle we took the question from.

Heres the question: 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. Twenty samples, each 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, whether or not there is evidence that the filter bed is failing to work properly.

In the book, they base the normal distribution on a derived poisson. They say the average amount of particles passing through the filtration bed is 50 and get this by finding the average amount of particles in 20 litres (20*10) then multiplying it by 0.25 (the proportion that is said to get through). They then approximate using a normal distribution defined by X-N(50,50) and then get an answer of 0.0282.Now what I did, was find the average of particles in 20 litres (200), and then model the entire thing as a binomial distribution using the 0.75 as a probability of a particle being removed. Defined X-B(200,0.75).I then approximated using a normal distribution and that's where my method and the books method diverged in terms of values. My derived normal distribution was X-N(50,37.5). As you can see, the variances differ. Therefore, this brought me to a slightly smaller value of Z(-2.20) and my final probability was 0.0139 (I flipped it around and said 136 particles did not get filtered then found P(X<136.5).

Both of our resultant probabilities were smaller than the specified 0.05 significance level therefore we both arrived at the same conclusion, that the filtration bed was not working as well as he thought.

Would I get penalised for taking a difference approach about the question and not getting the same answer? As far as I'm concerned my method is as legit as theirs.What do you guys think?

Thanks in advance.
I don't know whether you'd get penalised for it, but going from poisson to binomial is unconventional in S2 and a double approximation would lead to a pretty inaccurate, dubious answer. You'd definitely lose an accuracy mark in my opinion.
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Student7654
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(Original post by aymanzayedmannan)
I don't know whether you'd get penalised for it, but going from poisson to binomial is unconventional in S2 and a double approximation would lead to a pretty inaccurate, dubious answer. You'd definitely lose an accuracy mark in my opinion.
Was the modelling as a binomial really an approximation? I see it as me using the information I had given to me and manipulating it to get a feasible binomial model. Does that constitute an approximation?

I did the exact same as they did in the solution, but I used a binomial instead of a poisson as a starting point.
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Zacken
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(Original post by Student7654)
I did the exact same as they did in the solution, but I used a binomial instead of a poisson as a starting point.
Thats the problem, isn't it? It's a poisson model not a binomial. I'm not sure why you think it's feasibly binomial.
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TeeEm
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(Original post by Zacken)
Thats the problem, isn't it? It's a poisson model not a binomial. I'm not sure why you think it's feasibly binomial.
Need a bigger picture !!!
I cannot see your face, even with my reading glasses plus a magnifying glass!!
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tinkerbella~
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(Original post by TeeEm)
Need a bigger picture !!!
I cannot see your face, even with my reading glasses plus a magnifying glass!!
You don't want to see his face, trust me.
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TeeEm
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(Original post by tinkerbella~)
You don't want to see his face, trust me.
a girl with a moustache and a monobrow?!
good grief !!
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tinkerbella~
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(Original post by TeeEm)
a girl with a moustache and a monobrow?!
good grief !!
And still better looking than Zacken!
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Student7654
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(Original post by Zacken)
Thats the problem, isn't it? It's a poisson model not a binomial. I'm not sure why you think it's feasibly binomial.
Because there are 200 particles passing through the filter. Each one has a 0.75 chance of being filtered out since that's the filtration rate.. so can't you just model that as a binomial where each particle passing through is an independent trial with p=0.75? I don't understand why it wouldn't work.
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tiny hobbit
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(Original post by Student7654)
Because there are 200 particles passing through the filter..
No there aren't, there are between 0 and infinity (nearly), following a Poisson distribution with mean 200.
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Student7654
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(Original post by tiny hobbit)
No there aren't, there are between 0 and infinity (nearly), following a Poisson distribution with mean 200.
Okay so there is an average of 200 particles passing through, which is represented in the binomial distribution. I'm having a hard time understanding why they used poisson over binomial; whether its arbitrary or has a theoretical explanation. Sorry if I'm being difficult xD
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tiny hobbit
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(Original post by Student7654)
Okay so there is an average of 200 particles passing through, which is represented in the binomial distribution. I'm having a hard time understanding why they used poisson over binomial; whether its arbitrary or has a theoretical explanation. Sorry if I'm being difficult xD
Is this from the Edexcel book? If so, it reminds me as to why I always used my own questions for hypothesis testing. In their enthusiasm for finding "interesting" questions, they make up questions which cause confusion in the minds of those trying to answer them.

I would go with the theory, expressed by others, that if you start with a "story" that is Poisson, you don't go backwards to Binomial. Binomial relies upon having a fixed number of trials, which in this case would be the number of particles passing through. In this case it is not fixed, it varies with an average of 200.
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Student7654
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(Original post by tiny hobbit)
Is this from the Edexcel book? If so, it reminds me as to why I always used my own questions for hypothesis testing. In their enthusiasm for finding "interesting" questions, they make up questions which cause confusion in the minds of those trying to answer them.

I would go with the theory, expressed by others, that if you start with a "story" that is Poisson, you don't go backwards to Binomial. Binomial relies upon having a fixed number of trials, which in this case would be the number of particles passing through. In this case it is not fixed, it varies with an average of 200.
When i first saw the question, I saw a piece of poisson information and a piece of binomial information so I assumed it was viable to go either way. I asked my tutor on my next lesson and he told me the same thing as you just did, that binomial requires a fixed amount of trials and here the trials are indefinite and very roughly estimated. It was simply due to a lack of understanding about the conditions of a binomial.

It was an actual exam question by the way! It wasn't just one from the textbook.

Thanks for the reply
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