Confidence Intervals for Means
In the previous lesson, we learned about point estimates and margin of error. Now we combine these ideas into one of the most important tools in statistics: confidence intervals.
Confidence intervals help us estimate an unknown population mean using sample data, while clearly expressing uncertainty.
What Is a Confidence Interval?
A confidence interval is a range of values used to estimate a population parameter.
It is constructed using:
- A point estimate (sample mean)
- A margin of error
The general form is:
Confidence Interval = Sample Mean ± Margin of Error
What Does “Confidence” Mean?
The confidence level represents how confident we are that the interval captures the true population mean.
Common confidence levels are:
- 90%
- 95%
- 99%
A 95% confidence level means that if we repeat the sampling process many times, about 95% of the constructed intervals would contain the true mean.
Key Components of a Confidence Interval
| Component | Description |
|---|---|
| Sample Mean | Center of the interval |
| Margin of Error | Controls the width of the interval |
| Confidence Level | Indicates reliability of the interval |
Simple Numerical Example
A sample of 40 students has an average test score of 75.
Assume the margin of error is ±3.
The confidence interval is:
75 − 3 to 75 + 3 → (72, 78)
We are confident that the population mean lies between 72 and 78.
Effect of Confidence Level
Increasing the confidence level makes the interval wider. Decreasing it makes the interval narrower.
| Confidence Level | Interval Width |
|---|---|
| 90% | Narrower |
| 95% | Moderate |
| 99% | Wider |
Real-World Example
A delivery company estimates the average delivery time to be 4.2 days with a 95% confidence interval of (3.8, 4.6).
This means the company is reasonably confident that the true average delivery time lies within this range.
Important Interpretation Notes
- The confidence interval may or may not contain the true mean
- Confidence refers to the method, not a single interval
- A wider interval reflects greater uncertainty
Common Misunderstandings
- The true mean is not guaranteed to be inside the interval
- Higher confidence does not mean more precision
- Confidence intervals do not fix biased data
Quick Check
What happens to the confidence interval if confidence level increases?
Practice Quiz
Question 1:
What does a 95% confidence interval mean?
Question 2:
If the margin of error is 2 and the sample mean is 50,
what is the confidence interval?
Question 3:
Does increasing sample size generally widen or narrow the interval?
Mini Practice
A sample of products has an average weight of 500 grams with a margin of error of ±20 grams.
- What is the confidence interval?
- Would increasing the sample size reduce uncertainty?
What’s Next
In the next lesson, we will study Confidence Intervals for Proportions, which apply similar ideas to percentages and probabilities.