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

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

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

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

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

DEPARTMENT OF STATISTICS What proportion of NZ Year 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 =

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 pupils (mainly) run around at lunchtime? Sample (n = 30) Population 0.53 or 53%

DEPARTMENT OF STATISTICS What proportion of NZ Year 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...

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

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

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

DEPARTMENT OF STATISTICS Sampling variability Sample 1 (n = 30) 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

DEPARTMENT OF STATISTICS Sampling variability Sample 1 (n = 30) 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%

DEPARTMENT OF STATISTICS Sampling variability Sample 1 (n = 30) 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%

DEPARTMENT OF STATISTICS What proportion of NZ Year 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 =

DEPARTMENT OF STATISTICS What proportion of NZ Year 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%

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

DEPARTMENT OF STATISTICS

n = 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

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

DEPARTMENT OF STATISTICS

Different sample sizes n = 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%

DEPARTMENT OF STATISTICS Different sample sizes n = 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%

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 pupils (mainly) run around at lunchtime?

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 pupils (mainly) run around at lunchtime?

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 pupils (mainly) run around at lunchtime?

DEPARTMENT OF STATISTICS

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 pupils (mainly) run around at lunchtime?

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

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