What proportion of New Zealand Year 5–10 pupils

mainly run around in their lunchtime?

Population

My sample
(n = 30)

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

So, I estimate that about of all Yr 5–10 pupils mainly run around during their lunchtime (the population proportion).

What did others in the class get for their estimates?

I notice that:

(Go to the section headed ‘Sampling variability’ over the page.)

It is 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 away from my sample proportion of

somewhere between

with a margin of error of

I am giving a range of possible values for the population proportion rather than just using my single sample value. By doing this, I am taking account of the fact that if I had taken a different sample I would have a different sample proportion (sampling variability).

I notice that:

(Go to the section headed ‘Different sample sizes’’ over the page.)

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

250 if I want it to be a fairly safe bet that my sample proportion is no further than about 0.06 (6%) away from the population proportion

1000 if I want it to be a fairly safe bet that my sample proportion is no further than about 0.03 (3%) away from the population proportion.

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

My sample proportion =

With a sample of 1000 pupils, it is a fairly safe bet that my sample proportion is no further than away from the population proportion.

It is a fairly safe bet that between and of all Year 5–10 pupils mainly run around in their lunchtime.

Summary:
To find a population proportion:

I will take a random sample of n people from the population

I will find the proportion of interest in the sample, i.e., the sample proportion

I know about sampling variability (a different sample would give a different sample proportion) and I will account for sampling variability by giving a margin of error for my sample proportion and/or a range of possible values for the population proportion.

For a sample of size n, I will claim that:

my sample proportion is no further than away from the population proportion

my sample proportion estimates the population proportion with a margin of error of

the population proportion is somewhere between

my sample proportion – and my sample proportion +

Warning: This quick ‘’ method for finding the margin of error for a sample proportion works well when the sample proportion is somewhere between 30% and 70%. If the sample proportion is outside this range then using this method gives a margin of error which is far too big.

Exercise:

Find a survey which have been reported in a newspaper or via the internet.

Has the sample size and the margin of error been given?

Have they used this quick and easy rule for finding the margin of error?

See the Warning note above. Has the survey reported any sample proportions bigger than 70% or smaller than 30%? What could you say about the true margin of error for these sorts of sample proportions?

Sampling variability

Plot some of the proportions that others in the class got and then continue to take a lot more samples (all with n = 30) from the population and plot their sample proportions. Look at the pattern of variation in the sample proportions.

Population

… …

Sample 3
(n = 30) Sample 2
(n = 30) Sample 1
(n = 30) Sample
(n = 30)

Sample Proportions

I notice that:

Since sometimes the sample proportion will be bigger than the population proportion

and sometimes smaller than the population proportion, then in the long run,

the average (centre) of the sample proportions = the population proportion.

It is a fairly safe bet that:

any sample proportion will be no further than away from the population proportion

the population proportion will be no further than away from any sample proportion

(See section 3 on the front page now)

Different sample sizes

For n = 100, 250 and 1000 take lots of samples, plot the sample proportions on the axis and look at what happens to the pattern of variation in these sample proportions.

n = 100

Sample proportions

I notice that:

It is a fairly safe bet that the population proportion will be no further than
away from any sample proportion. (Note:)

n = 250

Sample proportions

I notice that:

It is a fairly safe bet that the population proportion will be no further than
away from any sample proportion. (Note: )

n = 1000

Sample proportions

I notice that:

It is a fairly safe bet that the population proportion will be no further than
away from any sample proportion. (Note: )

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.

(See section 4 on the front page now)