1. DEPARTMENT OF STATISTICS Statistical inference sampling variability estimate intervals sampling/ non sampling errors parameters estimates sample size effect margins of error variability of an estimate margin of error in the media A unit of work prepared and presented by: Marina Mcfarland & Anne Blundell (AGGS) Matt Regan (Stats Dept) Draft curriculum 2006

2. DEPARTMENT OF STATISTICS Draft curriculum 2006: Statistical investigation (thinking) Level 6: - making informal inferences about populations from sample data and communicating findings Level 7: - using sample statistics to make point estimates of population parameters - recognising the effect of sample size on the variability of an estimate Level 8: - determining estimates and confidence intervals for mean, proportions and differences

3. DEPARTMENT OF STATISTICS Level 7 - identifying sampling and possible non-sampling errors in polls and surveys Level 8 -... interpreting margins of error Draft curriculum 2006: Statistical investigation (thinking)

4. DEPARTMENT OF STATISTICS What proportion of NZ Year 5 -10 pupils (mainly) run around at lunchtime? Population Sample (n = 30)

5. DEPARTMENT OF STATISTICS What proportion of NZ Year 5 -10 students (mainly) run around at lunch time?

6. DEPARTMENT OF STATISTICS What proportion of NZ Year 5 -10 pupils (mainly) run around at lunchtime? Sample (n = 30) Population 0.53 or 53% 1.The proportion of pupils in my sample who mainly run around during their lunchtime =

7. DEPARTMENT OF STATISTICS 2.So, I estimate that about of all Yr 5–10 pupils mainly run around during their lunchtime. What proportion of NZ Year 5 -10 pupils (mainly) run around at lunchtime? Sample (n = 30) Population 0.53 or 53%

8. DEPARTMENT OF STATISTICS What proportion of NZ Year 5 -10 pupils (mainly) run around at lunchtime? What did others in the class get for their estimates? I notice that: Everyone seemed to get a different proportion for their sample There was a huge range of sample proportions...

9. DEPARTMENT OF STATISTICS Sampling variability Sample 1 (n = 30) 0.500.40.60.70.30.20.10.80.91 Sample 2 (n = 30) Sample 3 (n = 30) Sample Proportions

10. DEPARTMENT OF STATISTICS What proportion of NZ Year 5 -10 students (mainly) run around at lunch time?

11. DEPARTMENT OF STATISTICS Sampling variability Sample 1 (n = 30) 0.500.40.60.70.30.20.10.80.91 Sample 2 (n = 30) Sample 3 (n = 30)... Sample Proportions

12. DEPARTMENT OF STATISTICS Sampling variability Sample 1 (n = 30) 0.500.40.60.70.30.20.10.80.91 Sample 2 (n = 30) Sample 3 (n = 30)... Sample Proportions I notice that: Each sample has different people The sample proportions vary The variability in the sample proportions is large

13. DEPARTMENT OF STATISTICS Sampling variability Sample 1 (n = 30) 0.500.40.60.70.30.20.10.80.91 Sample 2 (n = 30) Sample 3 (n = 30)... Sample Proportions It is a fairly safe bet that: any sample proportion will be no further than 20% away from the population proportion ≈ 20%

14. DEPARTMENT OF STATISTICS Sampling variability Sample 1 (n = 30) 0.500.40.60.70.30.20.10.80.91 Sample 2 (n = 30) Sample 3 (n = 30)... Sample Proportions It is a fairly safe bet that: the population proportion will be no further than 20% away from any sample proportion ≈ 20%

15. DEPARTMENT OF STATISTICS What proportion of NZ Year 5 -10 pupils (mainly) run around at lunchtime? Sample (n = 30) Population 0.53 or 53% 1.The proportion of pupils in my sample who mainly run around during their lunchtime =

16. DEPARTMENT OF STATISTICS What proportion of NZ Year 5 -10 pupils (mainly) run around at lunchtime? I notice that: This range of possible values for the population proportion is so wide that it is practically useless. 3.It’s a fairly safe bet that the proportion of all Yr 5–10 pupils who mainly run around during their lunchtime (i.e., the population proportion) is: no further than 20% away from my sample proportion of 53% somewhere between 33% and 73% 53% with a margin of error of 20%

17. DEPARTMENT OF STATISTICS Different sample sizes 0.500.40.60.70.30.20.10.80.91 Sample 1 (n = 100)... Sample Proportions Sample 2 (n = 100)

18. DEPARTMENT OF STATISTICS

19. n = 100 0.5 0.45 0.550.60.40.350.30.650.7 Sample Proportions I notice that: It is a fairly safe bet that the population proportion will be no further than 9% away from any sample proportion. (Note: ) ≈ 9% Different sample sizes

20. DEPARTMENT OF STATISTICS Different sample sizes 0.500.40.60.70.30.20.10.80.91 Sample 1 (n = 250)... Sample Proportions Sample 2 (n = 250)

21. DEPARTMENT OF STATISTICS

22. Different sample sizes n = 250 0.5 0.45 0.550.60.40.350.30.650.7 Sample Proportions I notice that: It is a fairly safe bet that the population proportion will be no further than 6% away from any sample proportion. (Note: ) ≈ 6%

23. DEPARTMENT OF STATISTICS Different sample sizes n = 1000 0.5 0.45 0.550.60.40.350.30.650.7 Sample Proportions I notice that: It is a fairly safe bet that the population proportion will be no further than 4% away from any sample proportion. (Note: ) ≈ 4%

24. DEPARTMENT OF STATISTICS Conclusion: It is a fairly safe bet that for a random sample of size n, the population proportion will be no further than from the sample proportion. What proportion of NZ Year 5 -10 pupils (mainly) run around at lunchtime?

25. DEPARTMENT OF STATISTICS 4. When I use a sample proportion to estimate a population proportion then I should use a sample size, n, of at least: n = 250 if I want my sample proportion to be no further than about 0.06 or 6% (= ) away from the population proportion n = 1000 if I want my sample proportion to be no further than about 0.03 or 3% (= ) away from the population proportion What proportion of NZ Year 5 -10 pupils (mainly) run around at lunchtime?

26. DEPARTMENT OF STATISTICS 5. Randomly select a sample of size n = 1000 pupils from the population of all Yr 5–10 pupils: What proportion of NZ Year 5 -10 pupils (mainly) run around at lunchtime?

27. DEPARTMENT OF STATISTICS

28. It’s a fairly safe bet that between 48% and 54% of all Yr 5–10 pupils mainly run around in their lunchtime. 5. Randomly select a sample of size n = 1000 pupils from the population of all Yr 5–10 pupils: My sample proportion = 0.51 or 51% With 1000 pupils in my sample, it is a fairly safe bet that my sample proportion is no further than 0.03 or 3% away from the population proportion. What proportion of NZ Year 5 -10 pupils (mainly) run around at lunchtime?

29. DEPARTMENT OF STATISTICS What proportion of NZ Year 5 -10 pupils (mainly) run around at lunchtime? Summary Warning Exercise

30. DEPARTMENT OF STATISTICS Sampling / nonsampling errors ‘Random sample’ versus ‘simple random sample’ Population size and the margin of error Is ‘n = 30’ sufficiently large to satisfy underlying theory? Some issues