T-intervals

Math

Definition

A t-interval is a confidence interval for a population mean constructed using the t-distribution instead of the normal distribution, typically when the population standard deviation is unknown and the sample size is small. It gives a range of plausible values for the true population mean based on sample data.

How It Works

  1. Collect a random sample and calculate the sample mean and sample standard deviation.
  2. Determine the confidence level and find the corresponding t* critical value using degrees of freedom (n − 1).
  3. Calculate the standard error as s / √n.
  4. Construct the interval: x̄ ± t* × (s / √n).
  5. Interpret the interval in context of the problem.

Examples

  • Estimating the average test score of all students in a school from a sample of 25 students
  • Finding a 95% confidence interval for the mean weight of backpacks carried by middle schoolers
  • Determining a plausible range for the average commute time of employees at a company
Key Fact

t-interval formula: x̄ ± t* × (s / √n), where df = n − 1

Related Terms

One-sample t-test Confidence intervals for proportions Sampling distribution of a mean Central limit theorem

Study This Concept

Practice t-intervals with free review games in these units:

AP Statistics Unit 7: Inference for Means