Build and interpret a confidence interval for a population mean

Primary Tool

Confidence Interval Calculator

The confidence interval calculator takes sample mean, standard deviation, sample size, and $z$ -value and reports the interval directly. It also makes it easy to see how the interval width responds to changes in $n$ or the confidence level.

Problem

A sample of $n = 40$ students produces a mean exam score of $\overset{x}{ˉ} = 72.4$ with sample standard deviation $s = 8.3$ . Construct a 95% confidence interval for the population mean $μ$ .

This is a standard inference problem. The sample is large enough to use the $z$ -interval, and the numbers are chosen to make each step of the calculation clear.

Where the interval comes from

The sample mean $\overset{x}{ˉ}$ estimates $μ$ , but any single sample produces a slightly different $\overset{x}{ˉ}$ . The standard error $SE = s / n$ measures how much that variation is expected to be.

The 95% confidence interval is $\overset{x}{ˉ} \pm z^{*} \cdot SE$ , where $z^{*} = 1.96$ is the $z$ -score that puts 95% of the standard normal distribution between $- z^{*}$ and $z^{*}$ .

Step-by-step walkthrough

The four steps follow directly from the formula.

1Compute the standard error: $SE = s / n = 8.3/ 40 \approx 1.313$ .
2Identify the critical value for 95% confidence: $z^{*} = 1.96$ .
3Compute the margin of error: $ME = z^{*} \cdot SE = 1.96 \times 1.313 \approx 2.573$ .
4Build the interval: $(\overset{x}{ˉ} - ME, \overset{x}{ˉ} + ME) = (72.4 - 2.57, 72.4 + 2.57) = (69.83, 74.97)$ .
5State the result: the 95% confidence interval for the population mean is approximately $(69.83, 74.97)$ .

What the interval does and does not say

The correct interpretation is about the procedure, not this specific interval: if the same study were repeated many times with samples of the same size, 95% of the resulting intervals would contain the true $μ$ .

The true mean $μ$ is fixed — it does not have a probability of being in any interval. Either it is inside $(69.83, 74.97)$ or it is not. You cannot assign a 95% probability to that specific interval after observing the data.

A narrower interval is not automatically better. To get a narrower interval at the same confidence level you need a larger $n$ , which costs more data. Lowering the confidence level also narrows the interval but makes the coverage guarantee weaker.

Common Pitfall

A 95% confidence interval does not mean there is a 95% probability that the true mean lies in this particular interval. It means that the procedure that produced this interval captures the true mean in 95% of repeated samples. The distinction matters because $μ$ is a fixed number, not a random variable.

Try a Variation

Keep the same sample mean and standard deviation but double the sample size to $n = 80$ . How does the interval width change? What does that tell you about the relationship between sample size and precision?

Keep moving through the cluster

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