Skip to main content
MathidiaMathematics Academy
Statistical foundations
Lesson 43 of 75
Standard practice

Confidence intervals

Objective. Construct and interpret confidence intervals.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

A confidence interval is a range of plausible values for an unknown population parameter, constructed so that, across many repeated samples, a specified proportion (the confidence level) of such intervals would contain the true parameter. For a population mean with known standard deviation σ, a (1 − α)100% confidence interval is x̄ ± z_(α/2)(σ/√n), where z_(α/2) is the critical value from the standard normal distribution. Increasing the confidence level widens the interval, since greater certainty requires a broader range, while increasing sample size n narrows it by reducing the standard error. For a sample mean of 50 with σ = 10 and n = 25, a 95% confidence interval is 50 ± 1.96(10/√25), giving roughly 46.08 to 53.92.

Worked example

A sample of n = 36 has mean 20 and known standard deviation σ = 6. Construct the 95% confidence interval for the population mean.

3620695
Bar chart of the values 36, 20, 6, 95
  1. 1Identify the critical value for 95% confidence: z_(0.025) = 1.96.
  2. 2Compute the standard error: σ/√n = 6/√36 = 1.
  3. 3Multiply to find the margin of error: 1.96 × 1 = 1.96.
  4. 4Form the interval as the sample mean plus or minus this margin: 20 ± 1.96.

Answer: (18.04, 21.96)

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. Students interpret a 95% confidence interval as meaning there is a 95% probability the true parameter lies within this specific calculated interval; the correct interpretation refers to the long-run success rate of the method across repeated samples.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard