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S2 Chapter 5: Normal Approximations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 29 th September 2014

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1 2 3 ? ? ?

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? This chapter is about how we can approximate either the Binomial or Poisson Distribution using the Normal Distribution.

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Continuity Corrections We wish to approximate the Binomial and Poisson distributions using a Normal distribution. One problem is that Binomial/Poisson are discrete whereas the Normal distribution is continuous. We apply something called a continuity correction to approximate a discrete distribution using a continuous one. 3 4 5 6 7 8 9 10 Discrete: Continuous: ?

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Continuity Corrections 3 4 5 6 7 8 9 10 Discrete: Continuous: Notice the range has been enlarged by an extra 0.5. ? 3 4 5 6 7 8 9 10 Discrete: Continuous: ?

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Examples DiscreteContinuous ? ? ? ? ? ? ? ?

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To approximate a Binomial Distribution as a Normal Distribution, we just copy over the mean and variance of the Binomial to the Normal. Sorted! approximate ? ? Bro Tip: It’s a common error to accidentally forget to square root the variance to get the standard deviation – we need it in calculation of probabilities. ?

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Example Test Your Understanding: ? ? ?

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Exercise 5B 1 2 3 ? ? ? ? ? ? ?

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Again, we want the same mean and variance for the Normal as the original Poisson. approximate ? ?? Test Your Understanding ?

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Exam Example Edexcel S2 Jan 2012 (300 hits per hour is 50 hits per 10 mins) ? ?

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All Approximations Summarised! This handy diagram (found in your textbook) summarises all approximations very nicely. This will help you decide when the exam question says “Using a suitable approximation…” DiscreteContinuous Continuity correction needed if line crossed/ YES NO

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On the next few slides, read the question and vote with your diaries what the original distribution is and what the new approximated one is.

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Q: A spinner is designed to land on red 10% of the time. Use a suitable approximation to estimate the probability of fewer than 4 red in 60 turns of the spinner.

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Q: A spinner is designed to land on red 10% of the time. Use a suitable approximation to estimate the probability of more than 20 red in 150 turns of the spinner.

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Q: A Royal Mail processing centre receives on average 1200 letters a minute. Use a suitable approximation to work out the probability that it receives more than 1500 in a given minute.

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Exercises Edexcel S2 Jan 2013 Q2 Edexcel S2 May 2012 Q4 Edexcel S2 May 2011 Q5 ? ? ? ? ? ?

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