T-Test for Population Mean

Statistics · Axiom Academy

LESSON T-Test for Population Mean Understanding hypothesis testing when population standard deviation is unknown The t-test is used for hypothesis testing about a population mean when the population standard deviation (σ) is unknown , which is the most common real-world scenario. Population standard deviation σ is unknown We use sample standard deviation s as an estimate Sample comes from a normally distributed population (or n ≥ 30) The t-test statistic measures how many standard errors the sample mean is away from the hypothesized population mean. x̄ = sample mean (observed data) μ₀ = hypothesized population mean (null hypothesis) s = sample standard deviation (computed from data) s/√n = standard error of the mean Interpretation: The t-statistic tells us how far our sample mean is from μ₀, measured in units of standard error. Larger |t| values provide stronger evidence against the null hypothesis. The t-distribution's shape depends on the degrees of freedom (df), which accounts for the additional uncertainty from estimating σ with s. Effect of df: With smaller samples (lower df), the t-distribution has heavier tails than the normal distribution, reflecting greater uncertainty. As df increases, the t-distribution approaches the standard normal distribution. 4. Finding P-Values Using the T-Distribution The p-value represents the probability of observing a test statistic as extreme as (or more extreme than) the one calculated, assuming the null hypothesis is true.

This is the written version of the interactive lesson above. See the full Statistics course.