Confidence Interval Formula
The Formula
When to use: You can't know the exact average height of all Americans, but after measuring 200 people you can say: 'I'm 95\% confident the true average is between 167 cm and 173 cm.' It's like casting a net—wider nets catch the true value more often, but narrower nets are more useful. A 95\% confidence level means that if you repeated this process 100 times, about 95 of those nets would contain the true value.
Quick Example
Notation
What This Formula Means
A range of values, computed from sample data, that is likely to contain the true population parameter with a specified level of confidence.
You can't know the exact average height of all Americans, but after measuring 200 people you can say: 'I'm 95\% confident the true average is between 167 cm and 173 cm.' It's like casting a net—wider nets catch the true value more often, but narrower nets are more useful. A 95\% confidence level means that if you repeated this process 100 times, about 95 of those nets would contain the true value.
Formal View
Worked Examples
Example 1
mediumSolution
- 1 Standard error: SE = \frac{s}{\sqrt{n}} = \frac{16}{\sqrt{64}} = \frac{16}{8} = 2
- 2 Critical value for 95% CI: z^* = 1.96
- 3 Margin of error: E = z^* \times SE = 1.96 \times 2 = 3.92
- 4 95% CI: \bar{x} \pm E = 85 \pm 3.92 = (81.08, 88.92)
Answer
Example 2
hardCommon Mistakes
- Saying 'there is a 95\% probability the true mean is in this interval'—the true mean is fixed, not random; the interval is random.
- Forgetting that increasing sample size n narrows the interval (more precision), while increasing confidence level widens it (more certainty).
- Using a z-interval when the population SD is unknown and n is small—should use a t-interval instead.
Why This Formula Matters
Point estimates (like \bar{x} = 82) are almost certainly not exactly right. Confidence intervals honestly communicate how much uncertainty remains, which is essential for informed decision-making in polls, medical studies, and engineering.
Frequently Asked Questions
What is the Confidence Interval formula?
A range of values, computed from sample data, that is likely to contain the true population parameter with a specified level of confidence.
How do you use the Confidence Interval formula?
You can't know the exact average height of all Americans, but after measuring 200 people you can say: 'I'm 95\% confident the true average is between 167 cm and 173 cm.' It's like casting a net—wider nets catch the true value more often, but narrower nets are more useful. A 95\% confidence level means that if you repeated this process 100 times, about 95 of those nets would contain the true value.
What do the symbols mean in the Confidence Interval formula?
z^* is the critical value (e.g., 1.96 for 95\% confidence); s is the sample standard deviation.
Why is the Confidence Interval formula important in Math?
Point estimates (like \bar{x} = 82) are almost certainly not exactly right. Confidence intervals honestly communicate how much uncertainty remains, which is essential for informed decision-making in polls, medical studies, and engineering.
What do students get wrong about Confidence Interval?
A 95\% CI does NOT mean there's a 95\% probability the true parameter is in this specific interval. It means 95\% of similarly constructed intervals would contain the true parameter.
What should I learn before the Confidence Interval formula?
Before studying the Confidence Interval formula, you should understand: sampling distribution, central limit theorem, z score.