Sometimes it happens that we have have a list of data. We can calculate the mean easily, but the mean is specific to each sample. If we take another sample and calculate a new mean, the new mean and the old mean may be different. We might want to know how reliable our estimate of the mean is. Are we 90% confident? 95% confident? 99% confident?

To answer this we can find a confidence interval: if for example, we construct a 90% confidence interval, we can say, if we take many samples and find the mean of each one, then 90% of the time, the true mean will lie in the confidence interval.

If it is know that the underlying distribution is from a normal distribution, and we know the true or population standard deviation, then we can use the expression for a normal confidence interval:

where is the mean of the sample, is the sample size, and is the population standard deviation.

Suppose then we have the sample 2.3, 4.2, 5.3, 2.4, 2.6, 4.7 for lengths of french snails and we know that the lengths of snails are normally distributed with a standard deviation, of 1.7.

The mean of our sample is . We need to find the value of corresponding to a confidence interval of 90% or 0.9. This means a rejection region of area at the upper and lower ends. From the Normal tables, for The confidence interval is then

Suppose instead that we didn't know the population standard deviation, but we knew that the underlying distribution of the lengths was normal. We can can find an estimate for the standard deviation, called the sample standard deviation, from the original sample. We label it

Now though, since we estimated we must use the t-distribution with (n-1)=5 degrees of freedom, t_0.05=,5=2.015. The confidence interval is