Confidence Interval Interpretation
Imagine you're trying to guess the average height of all students in your school. You can't measure everyone, so you take a random sample of 100 students and find their average height is 165 cm. But you know that if you took a different random sample of 100 students, you'd probably get a slightly different average — maybe 164 cm, maybe 166 cm. The true school-wide average is somewhere around there, but you're not sure exactly where.
A confidence interval is a way of saying: "Based on my sample, I'm reasonably sure the true average lies somewhere between these two numbers." For example, you might calculate that the 95% confidence interval for the average height is (163 cm, 167 cm).
The most common mistake students make is thinking this means: "There is a 95% chance that the true average lies between 163 cm and 167 cm." That is wrong.
Why is it wrong? Because the true average is a fixed number — it either is between 163 and 167, or it isn't. There's no probability about it after you've calculated the interval. The probability was in the process before you collected the data.
The Correct Interpretation
Here's the precise statement: If you repeated this sampling process many, many times, and each time you calculated a 95% confidence interval, then about 95% of those intervals would contain the true population average.
Think of it like a fishing net. You cast your net (take a sample) and it lands somewhere in the ocean. The net covers a certain area (the confidence interval). The true fish (the population parameter) is somewhere in the ocean. You don't know if your particular net caught the fish. But you know that if you cast the net many times using the same method, about 95% of those casts would catch the fish.
The confidence is in the method, not in any single interval. A single interval either contains the true value or it doesn't — you just don't know which.
A Concrete Example
Suppose you're a quality control manager at a factory that produces 1-litre bottles of juice. You take a random sample of 50 bottles, measure their volumes, and calculate a 95% confidence interval for the true average volume: (0.995 L, 1.005 L).
What this means: If you took 100 different random samples of 50 bottles each, and calculated a 95% confidence interval from each sample, roughly 95 of those 100 intervals would contain the true average volume of all bottles. You have one of those intervals — you just don't know if yours is one of the 95 that work or one of the 5 that miss.
A helpful way to remember: The confidence level (95%) applies to the procedure, not to the interval. Say it aloud: "I am 95% confident that the procedure I used will produce an interval that contains the true value."
Why This Matters
In exams, you'll often be asked to interpret a confidence interval. The correct answer always involves the idea of repeated sampling. A typical exam answer would be:
"We are 95% confident that the true population mean lies between 163 cm and 167 cm." This means that if we were to take many samples and construct confidence intervals in the same way, about 95% of those intervals would contain the true mean.
Notice the careful wording: "We are 95% confident" — not "There is a 95% probability." The confidence is in the method, not in the specific numbers.