# AS 3.14 Apply Probability Distributions to Solve Problems

### Possible Lesson Sequence for AS 3.14 Distributions (AS 91586)

Distributions Starters (power point) (need to be updated 2013) Class site

1. Distributions Introduction - Shapes & Context: Normal, Binomial, Poisson, Uniform, Triangular, (True, Theoretical, Experimental models / past history) emphasis on applying (not just using a model)
2. Distribution Mean - review, estimation, effect. E(x), E(x + x + x ...), E(nx), E(x + y), E(x + A), E(x2)
3. Variance Var(x) using E(x2) - E(x)2   Var(x + x + x...), Var(nx) Var(x + y), Var(x + A) & Standard Deviation

4. Binomial Distribution - Overview and conditions / assumptions (new emphasis)
5. Binomial Distribution Graph - model vs actual, Mean & Standard Deviation
6. Binomial Tables
7. Binomial Tables when π > 0.5
8. Harder Binomial Problems
9. Binomial Review & Exam Questions

10. Poisson Distribution - introduction and formula - Overview and conditions / assumptions (new emphasis)
11. Poisson Distribution & model - graphs, mean, std dev, estimating λ
12. Poisson tables - adding, 1- (Mean & Standard Deviation) and changing λ
13. Practice problem solving
14. Inverse Poisson, Harder Poisson problems - trees, conditional
15. Poisson Review & Exam Questions

14. Uniform & Triangular Distributions (conditions / assumptions - new emphasis)
15. Practice of the Triangular Distribution

16. Normal Distribution
17. Inverse Normal distribution problems
28. Continuity Correction
29. Calculating mean & standard deviation / practice
20. More practice of Harder normal problems (combine with E(x) table, or trees or conditional)

21. Linear Combinations of Discrete distributions (Binomial & Poisson)
22. Combining skills - Trees, E(x), conditional, etc
23. True, Theoretical, Experimental models / past history) emphasis on applying (not just using a model) give a distribution graph - estimate the mean and std dev
24. Revision & Practice Test for AS 3.14 Distributions (or revision)

### For Teachers

* Be familiar with the Achievement Standard AS3.14 #91586 (link to NZQA)

* Check the latest information on the NZQA clarifications page as things change all the time, and the exemplars in NZQA may be out of date (as they are in this standard) Clarifications from NZQA Specifications Class notes

* Look over the NZQA Exemplars on AS3.14 Distributions Note: these exemplars may not reflect the latest information or the end of year exam... but we can only hope. Link to NZQA Sample Exam Paper | Schedule  Annotated Exemplars: Excellence | Merit | Achieve

* Material available on the mathstatsfacilitators web site AS 3.14 page

* AS3.14 page at CENSUS at schools site.

* Formula Sheet (link to NZQA)

* Check the latest comments on the NZmaths forum (login required)

* Past Examination Papers: (link to NZQA) 2013 (& schedule)

* Any further ideas / suggestions / resources/ comments / errors. Please contact maxr@nayland.school.nz

* AS 3.14 section on Nayland Web site updated: Sept 2014

AS 3.14 'Apply probability Distributions in solving problems' is a 4 credit externally assessed topic AS 91586

Summary:

Discrete random variables: Sums, differences, linear transformations
From an probability table: Expected value E(x) and Variance Var(x) and Standard Deviation
E(x), E(x + x + x ...), E(nx), E(x + y), E(x + A), Var(x) using the formula E(x2) - E(x)2   Var(x + x + x...), Var(nx) Var(x + y), Var(x + A) & Standard Deviation

Distributions: Uniform, Triangular, Normal, Poisson, (understanding of factorials & arrangements) Binomial, Inverse Normal, Inverse Poisson, Poisson changing λ,
Estimate mean & standard deviation from a distribution,
Conditions & Assumptions for applying the distribution

Combining Problems: Binomial or Poisson or normal with Expected value tables or probability trees,
Normal distribution combined with conditional probability.
Deciding what distribution to use to model a situation.
Continuity correction (when a continuous distribution is used to model a discrete variable OR for a rounded variable),
Distribution of true probabilities versus distribution of model estimates of probabilities versus distribution of experimental estimates of probabilities (discussion needed)

More detail on Achievement Objective page

### Web links for more material on AS 3.14 Distributions (AS 91586) Distributions Overview (and others not in L3 Stats) www.mathcaptain.com