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Draft curriculum 2006

This unit focuses on statistical inference and sampling variability, particularly regarding the proportion of New Zealand Year 5-10 pupils who run around at lunchtime. It guides students through making informal inferences about populations based on sample data, understanding the impact of sample size on variability, and interpreting confidence intervals and margins of error. Through hands-on practice and analysis of different sample sizes, students will learn the importance of estimating population parameters accurately and recognizing potential sampling and nonsampling errors in surveys.

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Draft curriculum 2006

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  1. Draft curriculum 2006 Statistical inference sampling variabilityestimateintervalssampling/nonsampling errors parametersestimatessamplesizeeffect marginsoferrorvariability 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)

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

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

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

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

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

  8. 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. Sample 2 (n = 30) Sample 3 (n = 30) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Sampling variability Sample 1 (n = 30) Sample Proportions

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

  11. Sample 2 (n = 30) Sample 3 (n = 30) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Sampling variability . . . Sample 1 (n = 30) Sample Proportions

  12. Sample 2 (n = 30) Sample 3 (n = 30) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Sampling variability . . . Sample 1 (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. Sample 2 (n = 30) Sample 3 (n = 30) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Sampling variability . . . Sample 1 (n = 30) Sample Proportions ≈ 20% • It is a fairly safe bet that: • any sample proportion will be no further than 20% away from the population proportion

  14. Sample 2 (n = 30) Sample 3 (n = 30) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Sampling variability . . . Sample 1 (n = 30) Sample Proportions ≈ 20% • It is a fairly safe bet that: • the population proportion will be no further than 20% away from any sample proportion

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

  16. What proportion of NZ Year 5-10 pupils (mainly) run around at lunchtime? • 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% I notice that: This range of possible values for the population proportion is so wide that it is practically useless.

  17. Sample 2 (n = 100) Sample 1 (n = 100) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Different sample sizes . . . Sample Proportions

  18. ≈ 9% Different sample sizes n = 100 0.3 0.35 0.4 0.5 0.55 0.6 0.65 0.7 0.45 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: )

  19. Sample 2 (n = 250) Sample 1 (n = 250) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Different sample sizes . . . Sample Proportions

  20. ≈ 6% Different sample sizes n = 250 0.3 0.35 0.4 0.5 0.55 0.6 0.65 0.7 0.45 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: )

  21. ≈ 4% Different sample sizes n = 1000 0.3 0.35 0.4 0.5 0.55 0.6 0.65 0.7 0.45 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: )

  22. What proportion of NZ Year 5-10 pupils (mainly) run around at lunchtime? 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.

  23. 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? • When I use a sample proportion to estimate a population proportion then I should use a sample size, n, of at least:

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

  25. What proportion of NZ Year 5-10 pupils (mainly) run around at 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. It’s a fairly safe bet that between 48% and 54% of all Yr 5–10 pupils mainly run around in their lunchtime.

  26. What proportion of NZ Year 5-10 pupils (mainly) run around at lunchtime? • Summary • Warning • Exercise

  27. Some issues • 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?

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