department of statistics statistical inference sampling variability estimate intervals sampling/ non...

DEPARTMENT OF STATISTICS

Statistical inference sampling variability

estimate intervals sampling/nonsampling 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


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


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)


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

Population

Sample (n = 30)


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



Sample (n = 30)

Population

0.53 or 53%

1. The proportion of pupils in my sample who mainly run around during their lunchtime =


2. So, I estimate that about of all Yr 5–10 pupils mainly run around during their lunchtime.


Sample (n = 30)

Population

0.53 or 53%



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

. . .


Sampling variability

Sample 1

(n = 30)

0.50 0.4 0.6 0.70.30.20.1 0.8 0.9 1

Sample 2 (n = 30)

Sample 3 (n = 30)

Sample Proportions


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



Sample 1

(n = 30)

0.50 0.4 0.6 0.70.30.20.1 0.8 0.9 1

Sample 2 (n = 30)

Sample 3 (n = 30)

. . .

Sample Proportions



Sample 1

(n = 30)

0.50 0.4 0.6 0.70.30.20.1 0.8 0.9 1

Sample 2 (n = 30)

Sample 3 (n = 30)

. . .

Sample Proportions

I notice that:Each sample has different peopleThe sample proportions varyThe variability in the sample proportions

is large



Sample 1

(n = 30)

0.50 0.4 0.6 0.70.30.20.1 0.8 0.9 1

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%



Sample 1

(n = 30)

0.50 0.4 0.6 0.70.30.20.1 0.8 0.9 1

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%



Sample (n = 30)

Population

0.53 or 53%

1. The proportion of pupils in my sample who mainly run around during their 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%


Different sample sizes

0.50 0.4 0.6 0.70.30.20.1 0.8 0.9 1

Sample 1 (n = 100)

. . .

Sample Proportions

Sample 2 (n = 100)


n = 100

0.50.45 0.55 0.60.40.350.3 0.65 0.7Sample 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%

%101.01001




0.50 0.4 0.6 0.70.30.20.1 0.8 0.9 1

Sample 1 (n = 250)

. . .

Sample Proportions

Sample 2 (n = 250)



n = 250


I notice that:


(Note: )

≈ 6%

%606.02501



n = 1000


I notice that:


(Note: )

≈ 4%

%303.010001


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.n1



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

2501

• n = 1000 if I want my sample proportion to be no further than about 0.03 or 3% (= ) away from the population proportion

10001



5. Randomly select a sample of size n = 1000 pupils from the population of all Yr 5–10 pupils:



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.




• Summary

• Warning

• Exercise


• 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

department of statistics statistical inference sampling variability estimate intervals sampling/ non...

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