Math · Introductory statistics · Concept
Confidence intervals: the t interval for a mean
A confidence interval gives a range of plausible values for an unknown population mean: the sample mean plus or minus a margin of error, a critical value times the standard error s/√n, so a larger sample size gives a narrower interval. When σ is unknown, the critical value comes from the t distribution with n − 1 degrees of freedom. The confidence level describes how often the method captures the true mean, not the chance that one computed interval is right.
Estimate plus or minus a margin
The interval is centered on the sample mean. Its half-width, the margin of error, is a critical value times the standard error.
The standard error
s/√n measures how much the sample mean varies from sample to sample. It shrinks as n grows: quadrupling the sample size halves it.
Why t and not z
Using s in place of the unknown σ adds uncertainty, so the t distribution has heavier tails than the normal. With n − 1 degrees of freedom, its critical values approach the normal value 1.96, for 95%, as n grows.
| Degrees of freedom | t* |
|---|---|
| 5 | 2.571 |
| 10 | 2.228 |
| 15 | 2.131 |
| 30 | 2.042 |
| 100 | 1.984 |
| Normal (z*) | 1.960 |
What 95% confidence means
If the sampling were repeated many times, about 95% of the intervals built this way would contain μ. A single computed interval either contains μ or does not; the 95% describes the method.
Conditions
The sample should be random and the observations independent. The t interval also assumes a roughly normal population; with larger samples it tolerates moderate skew, but outliers in a small sample can mislead.
Width and sample size
Higher confidence widens the interval, and a larger sample narrows it. To halve the margin of error, take four times as many observations.
Common mistakes
- Saying there is a 95% probability that μ lies in this particular interval.
- Using z* = 1.96 with a small sample and an estimated s: use t* with n − 1 degrees of freedom.
- Dividing s by n instead of √n.
- Treating the interval as the range that holds 95% of the individual observations.
Key terms
- Confidence interval
- A range of plausible values for a population parameter, from a method that captures the true value in a set share of samples, such as 95%. One computed interval either contains the parameter or doesn’t; the 95% describes the method.
- Confidence level
- How often the interval method captures the true parameter over many samples, such as 95%. A higher confidence level gives a wider interval from the same data.
- Margin of error
- The half-width of a confidence interval: a critical value times the standard error of the estimate. It covers sampling variation only, not bias from a poor sample or bad measurements.
- Standard error
- The standard deviation of an estimate, such as a sample mean, from sample to sample. For a mean it is estimated by s/√n, so larger samples give smaller standard errors.
- Student t distribution
- A bell-shaped distribution like the standard normal but with heavier tails, used when the population standard deviation is estimated from the sample. With more degrees of freedom it gets closer to the normal.
- Degrees of freedom
- The number of values free to vary once estimates have been fixed, such as n − 1 for one sample’s standard deviation. It sets the shape of the t, chi-square and F distributions.
- Critical value
- The cutoff from a reference distribution, such as z* = 1.96 for 95% confidence, that marks off a chosen tail area. It depends on the distribution, the tails used and any degrees of freedom.
Work through an example
A random sample of 16 packages has mean mass 52.3 g and standard deviation 6.4 g. Build a 95% confidence interval for the mean mass of all packages.
Build a 95% t interval for a mean →Sources and scope
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