One-Sample t-Interval
Construct and interpret a one-sample t-interval x̄ ± t*(s/√n). Understand why t replaces z when σ is unknown. Use the t-distribution with df = n−1 to find t* and interpret the resulting interval in context.
A lesson of AP Statistics, in Inference for Quantitative Data: Means, about 10 minutes.
One-Sample t Confidence Interval for a Mean
By the end of this lesson, you will be able to construct and interpret a one-sample t confidence interval for a population mean, check all required conditions, and determine the appropriate degrees of freedom.
When to Use a t Interval
Use a one-sample t interval when:
- You want to estimate a single population mean $\mu$
- The population standard deviation $\sigma$ is unknown (which is almost always)
- Data are from a random sample
The z interval (using z*) applies only when σ is known — a rare case in practice.
The t Distribution
The t distribution is like the standard normal but with heavier tails. It accounts for the additional uncertainty from estimating $\sigma$ with $s$.
Degrees of freedom (df) = n − 1. As df increases, the t distribution approaches the standard normal.
For a given confidence level, $t^* > z^*$ — the t distribution requires a wider interval to account for the extra uncertainty.
The Formula
$$\bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}}$$
where $t^*$ is the critical value from the t distribution with df = n − 1.
Margin of error = $t^* \cdot \dfrac{s}{\sqrt{n}}$
Conditions (STATE and CHECK on the AP Exam)
1. Random: Data must come from a random sample (SRS or random assignment).
2. Normal (10% / Normality condition):
- If n ≥ 30: CLT guarantees approximate normality (unless extreme outliers are present)
- If n < 30: The population should be approximately normal, or the sample data should show no strong skewness or outliers
- Check: dotplot or boxplot of the sample; look for severe departures from normality
3. Independent (10% condition): n < 10% of the population (ensures that sampling without replacement doesn't violate independence).
Step-by-Step Example
Context: A nutritionist wants to estimate the mean daily sodium intake of adults. A random sample of 25 adults shows $\bar{x} = 2850$ mg and $s = 420$ mg. Construct a 95% confidence interval.
State: We will construct a one-sample t interval for $\mu$ (mean daily sodium intake of adults).