Confidence Intervals

A confidence interval is a range of plausible values for a population parameter, built from sample data. It has the form point estimate ± margin of error. The confidence level, such as 95%, is how often the method captures the true value over many random samples.

Before this lesson: Sampling Distribution of a Sample Proportion, Sampling Distribution of a Sample Mean and Normal Probability Calculations

What a confidence interval means

A poll finds that 54% of 1,000 voters support a new law. The true share of all voters is probably not exactly 54%. A single number like 0.54 is a point estimate: our best guess, but almost surely a little off.

A confidence interval adds a cushion. The margin of error says how far the estimate is likely to be from the truth. A result like “54% ± 3%” gives the interval from 51% to 57%.

Why does this work? Sample results vary in a known way, as their sampling distribution shows. About 95% of sample proportions land within 1.96 standard deviations of the true p. So if you reach out 1.96 standard deviations from each sample result, about 95% of those intervals will contain p.

Confidence interval formula

\text{estimate} \pm \text{margin of error} = \text{estimate} \pm z^* \cdot \text{SE}

• The estimate is the sample statistic, such as p̂ or x̄.
• The critical value z* sets the confidence level. It is the number of standard deviations you reach out.
• SE is the standard error: the standard deviation of the statistic, estimated from the sample.

How to interpret a confidence interval

Use this template: “We are 95% confident that the interval from ___ to ___ captures the true [parameter in context].” For the confidence level, say: “If we took many random samples and built an interval from each, about 95% of the intervals would capture the true value.”

Calculator: to find z* on a TI-84, use invNorm(left area, 0, 1), where the left area is the confidence level plus one tail. In Desmos, use normaldist(0, 1).inversecdf(left area).

Confidence interval examples

Example 1: Build an interval from an estimate

A poll of 1,000 random voters finds p̂ = 0.54 with a margin of error of 0.03 at 95% confidence. Find and interpret the interval.

0.54 - 0.03 = 0.51

0.54 + 0.03 = 0.57

Answer: (0.51, 0.57). We are 95% confident that the interval from 51% to 57% captures the true share of all voters who support the law.

Example 2: Find the estimate and margin of error

A study reports the 95% confidence interval (12.4, 15.0) for the mean number of hours teens spend online each week. Find the point estimate and the margin of error.

12.4 + 15.0 = 27.4

\dfrac{27.4}{2} = 13.7

15.0 - 12.4 = 2.6

\dfrac{2.6}{2} = 1.3

Answer: the point estimate is x̄ = 13.7 hours, and the margin of error is 1.3 hours.

Example 3: Find a critical value

Find z* for an 80% confidence interval.

1 - 0.80 = 0.20

\dfrac{0.20}{2} = 0.10

0.80 + 0.10 = 0.90

z^* = \text{invNorm}(0.90) \approx 1.282

Answer: z* ≈ 1.282. The z-table gives about 1.28, since P(Z \lt 1.28) = 0.8997.

Example 4: Build an interval from a standard error

A random sample of adult men has mean height x̄ = 68.5 inches, with a standard error of 1.2 inches. Build a 95% confidence interval for the mean height of all adult men.

\text{ME} = 1.96(1.2)

= 2.352

68.5 - 2.352 = 66.148

68.5 + 2.352 = 70.852

Answer: about (66.15, 70.85). We are 95% confident that the interval from 66.15 to 70.85 inches captures the true mean height of adult men.

Example 5: Use an interval to judge a claim (word problem)

A health survey gives a 95% confidence interval of (0.28, 0.36) for the proportion of teens who sleep at least 8 hours on school nights. (a) Find the estimate and margin of error. (b) A website claims that 40% of teens sleep that much. Is the claim plausible? (c) Does the interval mean that 95% of individual teens fall between 0.28 and 0.36?

\dfrac{0.28 + 0.36}{2} = \dfrac{0.64}{2}

= 0.32

\dfrac{0.36 - 0.28}{2} = \dfrac{0.08}{2}

= 0.04

0.40 \gt 0.36

Answer: (a) p̂ = 0.32, with margin of error 0.04. (b) No. 0.40 is outside the interval, so 40% is not a plausible value. (c) No. The interval estimates one number, the proportion of all teens who sleep at least 8 hours. It says nothing about individual teens.

Common mistakes

Related methods

What changes the width

The margin of error is z* times SE. A higher confidence level makes z* larger, so the interval widens. A larger sample makes SE smaller, so the interval narrows. Because SE has √n in the denominator, you need 4 times the sample size to cut the margin of error in half.

Specific intervals

The next lessons build the confidence interval for a proportion, using p̂ and \sqrt{\hat{p}(1 - \hat{p})/n}, and for a mean (see confidence interval for a mean), using x̄ and a t critical value. Both rest on the central limit theorem and the conditions you check for sampling distributions.

Practice problems

1. A poll finds p̂ = 0.62 with margin of error 0.04. Write the confidence interval.
2. A confidence interval is (0.41, 0.49). Find the point estimate and the margin of error.
3. A confidence interval for a mean is (22.6, 27.0). Find the point estimate and the margin of error.
4. Find z* for a 99% confidence interval.
5. A sample has x̄ = 40 with standard error 2.5. Build a 90% confidence interval for μ.
6. What happens to the width of an interval when the confidence level changes from 95% to 99%?
7. The sample size is multiplied by 4, and the confidence level stays the same. What happens to the margin of error?
8. A student says, “There is a 95% probability that μ is between 12 and 18.” Explain the mistake.
9. A 95% interval for a proportion is (0.28, 0.36). Is 0.30 a plausible value of p?
10. Fifty different random samples each give a 95% confidence interval. About how many of the intervals should capture the true parameter?

0.62 - 0.04 = 0.58

0.62 + 0.04 = 0.66

\dfrac{0.41 + 0.49}{2} = \dfrac{0.90}{2}

= 0.45

\dfrac{0.49 - 0.41}{2} = \dfrac{0.08}{2}

= 0.04

\dfrac{22.6 + 27.0}{2} = \dfrac{49.6}{2}

= 24.8

\dfrac{27.0 - 22.6}{2} = \dfrac{4.4}{2}

= 2.2

1 - 0.99 = 0.01

\dfrac{0.01}{2} = 0.005

0.99 + 0.005 = 0.995

z^* = \text{invNorm}(0.995) \approx 2.576

\text{ME} = 1.645(2.5) = 4.1125

40 - 4.1125 = 35.8875

40 + 4.1125 = 44.1125

2.576 \gt 1.96

\dfrac{\sigma}{\sqrt{4n}} = \dfrac{\sigma}{2\sqrt{n}}

0.28 \lt 0.30 \lt 0.36

0.95(50) = 47.5

Confidence interval FAQ

What does a 95% confidence interval mean?

If you took many random samples and built a 95% interval from each one, about 95% of the intervals would capture the true parameter. For one interval, you say you are 95% confident it captures the parameter.

What is the margin of error?

It is the critical value times the standard error. It is half the width of the interval and shows how far the estimate is likely to be from the true value.

Why is 1.96 used for 95% confidence?

In a standard normal distribution, 95% of the area lies between z = −1.96 and z = 1.96. So 1.96 standard errors on each side captures the middle 95%.

Does a wider interval mean more confidence?

For the same data, yes. A higher confidence level uses a larger critical value, which makes the interval wider. More confidence costs precision.

Summary

• A confidence interval is estimate ± margin of error, where the margin of error is z* times SE.
• The estimate is the midpoint of the interval, and the margin of error is half its width.
• The confidence level is the long-run capture rate of the method, not the chance that one interval is right.
• Interpret in context: “We are 95% confident that the interval from ___ to ___ captures the true ___.”
• Higher confidence widens the interval; a larger sample narrows it.

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Next lesson: Confidence Interval for a Proportion

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