TL;DR:
Name the parameter in context, then use the template: we are
95%confident that the interval from a to b captures the true mean of the population.The confidence level describes the method, not the chance that this one interval is right.
Check the conditions (random sample, independence, approximately normal) before you calculate.
Computing a confidence interval takes a few lines. Explaining it is where marks and understanding are lost. Students write "there is a 95% chance the mean is in the interval", which sounds right and is not. The true mean is a fixed number: either it is in your interval or it is not.
This article computes an interval and then writes the sentence carefully, so you can reuse the template.
What the interval means
A confidence interval is built from a sample by a method. If you repeated the sampling many times and built an interval each time, a certain proportion of those intervals would contain the true parameter. That proportion is the confidence level. It is a statement about the method over many samples, not about the one interval you have.
So a 95% confidence level means the method captures the true value in about 95 out of 100 intervals built in the long run. It does not say that 95% of the data lie in the interval, and it does not say the mean has a 95% chance of being inside this one.
The steps
Name the parameter: Say what is being estimated and for which population: "the mean commute time of all students at the school".
Check the conditions: Random sample, a sample small compared with the population, and an approximately normal sample or a large enough size.
Compute the interval: Estimate ± critical value × standard error.
State the interpretation: Use the template, in the context of the problem, with units.
Say what the level means, if asked: One sentence about repeated sampling.
Worked example: a confidence interval for a mean
A random sample of 25 students from a school reports a mean commute of x̄ = 18.4 minutes with a sample standard deviation s = 6.0 minutes. The sample is small, and a plot shows no strong skew or outliers. Find a 95% confidence interval for the mean commute time of all students at the school.
Conditions. Random sample: given. Independence: the 25 students are a small part of the whole school, which is what the independence check asks for. Normality: with n = 25, the plot shows no strong skew or outliers, so a t procedure is reasonable.
Standard error.
SE = s / √n = 6.0 / √25 = 6.0 / 5 = 1.2.Critical value. For
95%confidence withdf = 25 − 1 = 24,t* ≈ 2.064from a t table.Margin of error.
ME = t* × SE = 2.064 × 1.2 = 2.48.Interval.
18.4 ± 2.48, which is(15.92, 20.88), or about(15.9, 20.9)minutes.
Interpretation. Use this template, with the numbers and context of your problem:
We are 95% confident that the interval from 15.9 to 20.9 minutes
captures the true mean commute time for all students at the school.The sentence names the level, gives the interval with units, and says what is being estimated and for whom. That is all the answer needs.
What the level means. If we took many random samples of 25 students and built an interval from each in this way, about 95% of those intervals would contain the true mean commute time.
Sense check. The interval is centred on the sample mean: (15.92 + 20.88) / 2 = 18.4. Its half-width is (20.88 − 15.92) / 2 = 2.48, equal to the margin of error.
What the interval does not say
It does not say that
95%of students have commutes between 15.9 and 20.9 minutes. It is about the mean, not individual students.It does not say the sample mean is in the interval. The sample mean is always in it.
It does not say there is a
95%probability for this particular interval, because the true mean is fixed.
Choosing the right procedure
Before computing, ask what you are estimating. A mean of a quantitative variable from one sample uses a one-sample t interval, as here. A proportion uses a different interval, and a difference between two groups uses another. Naming the procedure and checking its conditions is usually the first thing a reader looks for, so write it down even if the question does not ask.
If the question gives you software output, the work is mostly in reading it: find the interval, the sample size and the standard error, then write the same interpretation sentence. The sentence never changes shape, only the parameter and the numbers.
How width changes
A larger sample makes the standard error smaller, so the interval is narrower. A higher confidence level uses a larger critical value, so the interval is wider. For example, with n = 100 and the same s, SE = 6.0 / 10 = 0.6, which is half of the 1.2 we had for n = 25. The sample is four times larger and the square root of 4 is 2.
Mistakes to avoid
Saying there is a high probability that the true mean lies in this particular interval.
Leaving out the context or the units.
Skipping the conditions.
Using z when the population standard deviation is unknown and the sample is small. Use t.
Interpreting an interval for a mean as if it described individuals.
Your plan for this week
Write the template sentence on a card.
Compute two intervals from a textbook data set, checking conditions each time.
Write the interpretation and the meaning of the level for both.
Ask a friend to read them and spot any wording that makes a claim about probability.
Frequently asked questions
Can I say the true mean is probably in my interval?
In casual speech people do, but for a class answer use the confidence wording. It is the method that is described by the level.
Why use t instead of z?
The population standard deviation is usually unknown, so we estimate it with s, and the t distribution accounts for the extra uncertainty, especially in small samples.
Get started
