2.9

# Interpreting confidence intervals

Confidence intervals can give us an indication of how reliable our sample’s mean is.

By now, you should be able to use the mean and standard deviation of normally distributed data to make predictions about how likely you are to get a particular result. You could use these calculations to predict how often a bus will take longer than usual to make a stop and how much longer it will take. This is, of course, useful to know if you’re planning a public transport system.

The only unresolved issue we still have, if we’re trying to make an estimate, is whether or not we can be sure that the mean of our sample is close to the mean of our population. We can never know for certain, but we can calculate how confident we are that we would get a similar mean if we took another sample.

This calculation is called a confidence interval, and the results give us a range within which the population’s mean might plausibly fall.

Imagine a researcher takes a sample of how long it takes to complete a bus stop at a particular stop on 50 occasions. They may report a 95% confidence interval of 8.5 to 9.76 seconds for the mean duration of a bus stop. This means that the authors are 95% sure the true average duration of a bus stop falls within those times.

A confidence interval of 95% means that if you

• sampled bus stop durations many times and
• calculated the confidence interval for each sample, then
• the true population mean would fall within the range set out by these intervals approximately 95% of the time.

Practically speaking, this would mean we could be fairly confident that our bus stop times will be accurate to within one and a half seconds (or just under) when we factor stop times into the bus schedule.

Because larger samples give us more information to go on, confidence intervals become more precise as sample size grows.

### Why use a confidence interval?

The table below explains three common applications of a confidence interval.

Application of confidence interval Explanation
Indicate level of certainty that your mean represents the population’s mean Because so many statistics that we use to make predictions about what is likely to happen flow on from the mean, confidence intervals give us some indication of how confident we should be of our results.
Estimate required sample size Confidence intervals can be used to estimate the sample size you’ll need (Kadam and Bhalerao 2010). There are online calculators that can help you with this.
Indicate external validity If your sample’s confidence interval does not overlap with another sample’s, you may not be sampling from the same target population.

### Calculating intervals

Confidence intervals can be calculated manually without too much trouble once you’ve done a few, but there are also online calculators that will do the work for you if you’re in a hurry.

Over the last few steps you’ve looked at averages, normal distribution, standard deviation and confidence intervals.

In your own words explain how you can use these tools together to make predictions based on the data in your sample.

What are the potential applications of these predictions? Illustrate your response with examples.

If you like a challenge, apply these concepts to make predictions about the how much rain you are likely to have in January, April, July and November in your local area within a 95% confidence interval.

If you’re unable to find data on rainfall in your local area, use this data from Bungaree in Victoria, Australia instead.