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# S2 question 12, exercise 7D. watch

1. 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?

Why not try posting in a specific subject forum- you might have more luck there.

Here's a link to our subject forum which should help get you more responses.

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Updated: February 8, 2016
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