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

2. 1 2 3 ? ? ?

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

4. 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: ?

5. 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: ?

6. Examples DiscreteContinuous ? ? ? ? ? ? ? ?

7. 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. ?

8. Example Test Your Understanding: ? ? ?

9. Exercise 5B 1 2 3 ? ? ? ? ? ? ?

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

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

12. 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

13. 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.

14. 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.

15. 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.

16. 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.

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