Understanding and calculating confidence intervals for proportions.
Hi I'm Dr Nic from Creative Maths including Statistics Learning Centre.
In this video I'm going to explain what confidence intervals for proportions tell us,
how to calculate them, what the effect of different confidence levels is,
and some important things to remember when reporting confidence intervals.
To find out about confidence intervals in general,
watch our video, Understanding Confidence Intervals.
There are two main types of data - Categorial
and measurement.
With measurement data we usually summarise using the mean or median,
with a confidence interval giving an estimate of the range
within which we are pretty sure that that population parameter lies.
With categorical data, we summarise by giving
the proportions of responses or outcomes for each of the possible values.
It is quite common to see confidence intervals for proportions for political polls.
We cannot usually ask the entire population their opinion on a certain matter, or who
they would vote for. So we ask a sample of the people and draw
conclusions about the population from that sample.
We use a confidence interval to report the range within which we are pretty sure that
the population parameter lies.
Here is an example
We are exploring the idea of a dessert takeaway restaurant.
and we are interested to know people's habits around fast food.
We ask a sample of 153 people in New Zealand what their favourite takeaway food is.
Out of them, twenty say Fish and chips.
The sample proportion is 20 divided by 153,
which gives 0.13 or 13%.
We calculate the confidence interval for the population proportion
using this formula.
The standard error in this instance is the
square root of 0.13 times 0.87 divided by 153, which comes to 0.027.
This standard error is multiplied by the Z value, which in this case is 1.96.
We get a margin of error of 1.96 times 0.027 = 0.0529
This margin of error is added to and subtracted from the sample proportion to get a confidence
interval of 0.077 to 0.183. We can be 95% confident that between 8% and
18% of people in the population like fish and chips the best.
This is a pretty big interval, but that is because we had quite a small sample.
When samples are bigger, the width of our confidence interval reduces.
If we wished to be more certain to have the true population value within our confidence
interval, we will increase the stated confidence level.
This changes the Z value that we use. You can see in this table, that as we increase
our confidence level, from 90 to 95 to 99 percent
the width of the confidence interval increases.
The more confident we wish to be, the larger our confidence interval will be.
There are two conditions we need to test for, to make sure it is valid to use this formula.
We need to make sure we are not testing more than 10% of the population
And the following must hold: np>10 and n(1-p) >10.
This is because the formula works on the assumption
that the sampling distribution of the proportion is well-modelled by the normal distribution.
The sample size of 153 people is less than 10% of the adult population of New Zealand,
so we are fine there. np = 20 and n(1-p) = 133.
Both these values are greater than ten, so we are fine there as well.
Important things to remember:
This confidence interval gives us a range of values that we can be pretty sure (or confident)
contains the proprtion from the population. It is quite possible that this interval will
NOT contain the population proportion, but usually a confidence interval will contain
the population parameter we are estimating.
When we report a confidence interval we must be careful to say that the confidence interval
is about the population proportion. We KNOW what the sample proportion is, so we do not
need to make a confidence interval about it.
If you would like to know more, please ask
questions in the comments below or on the channel community chat.
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And may your confidence intervals be small. ;)