# Confidence intervals

### Confidence intervals

A confidence interval describes a range of values that are likely to include the true value for a population. The upper and lower confidence limits are the two numbers that make up the range of the interval. Confidence intervals do not provide certain answers, they are an estimate based on a sample.

Commonly either a 95% or 99% confidence interval is used in experiments, which equates to either a 5% or 1% significance level (p-value). The interval describes all values for which we cannot reject the null hypothesis at the given significance level. If a 95% confidence interval has been calculated and your experiment provides a value inside the interval, you can state that your conclusions would be right 95% of the time.

Here is an example of how to find a confidence interval for a population mean, which is relevant to both z-tests and t-tests. In this example:
n (number of observations) = 70
The sample standard mean (x-bar) = 48
The sample standard deviation = 15
We want to find a 99% confidence interval
The critical value for a z-test at 99% significance = 2.58

• Step 1 – Calculate the Estimated Standard Error (ESE), this is the square root of the square of the sample mean divided by n and can be calculated with an Excel formula: =SQRT(POWER(15,2)/70) which equals an ESE of 1.7928
• Step 2 – Calculate the lower limit as the sample mean minus the critical value multiplied by ESE, in Excel for this experiment this is =48-(2.58*1.7928) = 43.37
• Step 3 – Calculate the upper limit as the sample mean plus the critical value multiplied by ESE, as an Excel formula this is =48+(2.58*1.7928) = 52.63
• Step 4 – Your 99% confidence interval: (43.37, 52.63)
•

Note that the calculations are the same for a 95% confidence interval, apart from the critical value being 1.96 rather than 2.58. For 95% confidence, the interval is (44.49, 51.51).

There are Excel functions to speed up your confidence interval calculations, CONFIDENCE.NORM returns the confidence interval for a population mean based on the normal distribution and CONFIDENCE.T based on the t-distribution.

### Interpreting a confidence interval

Taking a random sample and applying a 95% confidence interval, the interval will give all values for the population mean that would not be rejected at the 5% significance level. Therefore on about 95% of occasions the true population mean is within the interval, however 5% of the possible random samples you could select will provide a confidence interval which does not include the population mean.

### Point estimate

A single number calculated from the data set, that is the best single guess for an unknown parameter. Point estimates can be deduced from a confidence interval by taking the midpoint of the interval.

### Interval estimate

A range of numbers around the point estimate, within which an unknown parameter is expected to fall. The confidence interval is a widely used type of interval estimate.