Understanding Confidence Intervals
In order to understand confidence intervals, we need to understand
sampling and sampling error.
To find things that about a population of interest,
it is common practice to take a sample.
A sample is a selection of objects or observations taken from the population of interest.
For example, a population might be all apples in an orchard at a given time.
We wish to know how big the apples are.
We can't measure all of them so we take a sample of some of them and measure them.
To find out about different sampling methods,
see our video, "Sampling: Simple, Random, Convenience, etc."
Inference is when we draw conclusions about the population from the sample.
Because the sample was only a selection of objects from the population, it will
never be a perfect representation of the population.
Different samples of the same population will give different results.
This is called sampling error or variation due to sampling.
There will always be sampling error.
Confidence Intervals
When we express an estimate of a population parameter,
it is good practice to give it as a confidence interval.
A confidence interval communicates how accurate our estimate is likely to be.
Say we wish to find out how big the apples are in our orchard.
We put this as an investigative question:
What is the main weight of all the apples in the orchard?
We take a sample, and calculate the sample mean.
This is the best estimate of the population mean.
We use a confidence interval to express the range in which we are pretty sure
the population parameter lies.
In this case the population parameter is the mean weight for all the apples in the orchard.
The width of a confidence interval depends on two things:
The variation within the population of interest,
and the size of the sample.
If all the values in the population were almost the same,
then our sample will also have little variation.
Any sample we take is likely to be pretty similar to any other sample.
Our estimate is going to be pretty close to the true population value.
We would have a small confidence interval.
But a more varied population will lead to a more varied sample.
Different samples taken of the same population will differ more.
We would be less sure that the sample mean was close to the population mean.
Our confidence interval would be larger.
So, greater variation in the population leads to a wider confidence interval.
Sample size also affects the width of a confidence interval.
If we take a small sample, we don't have much information on which to base
our inference.
Small samples will vary more from each other.
There is more variation due to sampling, or sampling error,
with a small sample.
In larger samples, the effect of a few unusual values is evened out by the other
values in the sample.
Larger samples will be more similar to each other.
The effect of sampling error is reduced with larger samples.
When we take a large sample,
We have more information and can be more sure about our estimate.
The confidence interval can be smaller.
There are several methods for calculating confidence intervals:
When we use traditional confidence interval formulas, the stated level
of confidence also effects the width of the confidence interval.
All estimates of population parameters, such as means, medians,
differences of means and differences in medians
should be expressed as confidence intervals.
You can learn more about how to calculate confidence intervals in our other videos.