Overview
This lecture introduces the T statistic and the T distribution, explaining their use in estimating and testing population means for numerical data, and highlighting how they differ from earlier methods used for categorical data.
Standard Error and the T Statistic
- Standard error formulas differ for proportions (categorical data) and means (numerical data).
- For means, standard error uses the sample standard deviation (s), not the population standard deviation (σ).
- The T statistic formula uses the sample mean (x̄), sample standard deviation (s), and sample size (n).
- The T statistic is based on the z-score formula but applies sample data throughout.
The T Distribution
- The T distribution is used for numerical data and population means when σ is unknown.
- It is symmetric and unimodal, similar to the normal distribution, but has thicker tails.
- Thicker tails mean the T distribution allows more extreme values, reflecting higher variability in numerical data.
- The shape of the T distribution depends on sample size, described by degrees of freedom (df = n - 1).
- As sample size increases, the T distribution approaches the normal distribution.
Implications for Inference
- For small sample sizes, use the T distribution for statistical inference about means.
- For large sample sizes, the T distribution becomes nearly normal.
- Statistical inferences like confidence intervals and hypothesis tests for means use the T statistic.
- Calculator functions for these procedures will start with "T" (e.g., T interval, T test).
Key Terms & Definitions
- T statistic — A standardized value calculated using the sample mean, sample standard deviation, and sample size.
- T distribution — A probability distribution used for inference about population means when the sample size is small and σ is unknown.
- Degrees of freedom (df) — Calculated as sample size minus one (n - 1); determines the shape of the T distribution.
- Standard error (for means) — An estimate of spread using sample standard deviation divided by the square root of n.
Action Items / Next Steps
- Prepare to use "T" functions on your calculator in sections 9.3 (confidence intervals) and 9.4 (hypothesis testing).
- Review how to compute the T statistic and interpret degrees of freedom.