Confidence Interval Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Confidence Interval.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

A range of values, calculated from sample data, that is likely to contain the true population parameter with a specified level of confidence.

Instead of saying 'the average is 50,' you say 'I'm 95% confident the average is between 47 and 53.' The interval acknowledges uncertainty from sampling.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A confidence interval gives a range of plausible values for a population parameter, constructed so that the procedure captures the true parameter a fixed percentage of the time.

Common stuck point: Students say '95% probability the true mean is in this interval.' That is wrong. The true mean is fixed; it is the interval construction process that is 95% reliable.

Worked Examples

Example 1

hard
A sample of 100 students has a mean test score of \bar{x} = 72 with population standard deviation \sigma = 10. Construct a 95% confidence interval for the population mean.

Solution

  1. 1
    Step 1: For 95% confidence, z^* = 1.96.
  2. 2
    Step 2: Standard error: \text{SE} = \frac{10}{\sqrt{100}} = 1.
  3. 3
    Step 3: CI = \bar{x} \pm z^* \cdot \text{SE} = 72 \pm 1.96(1) = (70.04, 73.96).

Answer

(70.04, 73.96)
A 95% confidence interval means that if we repeated this sampling process many times, about 95% of the intervals constructed would contain the true population mean.

Example 2

hard
A 95% confidence interval for the mean weight of apples is (150g, 170g). Interpret this interval.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

hard
A sample of 64 has \bar{x} = 50 and \sigma = 8. Find the 99% confidence interval (z^* = 2.576).

Example 2

hard
A 90% confidence interval for a population mean is given as 68 to 74. What are the sample mean and the margin of error?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

standard errorsampling distribution