One-Sample t-Test

One-Sample t-Test for Means

By the end of this lesson, you will be able to conduct a one-sample t-test, interpret t-statistics and p-values, construct confidence intervals for a mean, and check the necessary conditions.

When to Use the t-Test

Use the one-sample t-test when:

• Testing a claim about a population mean (μ)
• The population standard deviation (σ) is unknown (typical in practice)
• Either the sample is large (n ≥ 30) or the population is approximately normal

The t-test uses sample standard deviation (s) instead of σ, which introduces additional uncertainty captured by the t-distribution.

The t-Distribution

The t-distribution is bell-shaped and symmetric like the normal distribution, but with heavier tails — reflecting additional uncertainty from estimating σ with s.

Degrees of freedom (df): For one-sample t-tests, df = n - 1.

As df increases (larger sample), the t-distribution approaches the standard normal (z) distribution.

Key property: For the same significance level, the critical t-value is LARGER than the critical z-value — harder to achieve significance with a t-test.

Conditions for the t-Test

Before conducting a t-test, verify:

1. Random: The data comes from a random sample (not a convenience or voluntary sample).

2. Independent: Observations are independent of each other. If sampling without replacement from a finite population, sample size ≤ 10% of population.

3. Normal/Large sample:

• Either n ≥ 30 (Central Limit Theorem applies)
• OR the sample distribution shows no strong skewness or outliers (use dotplot, histogram, or boxplot)

If the distribution is severely skewed and n < 30, the t-test is not appropriate.

The Hypothesis Test: Step-by-Step

Step 1: State Hypotheses

Null hypothesis (H₀): μ = μ₀ (the population mean equals the claimed value)

Alternative hypothesis (Hₐ): Choose based on the question:

• Hₐ: μ ≠ μ₀ (two-sided: the mean differs from μ₀)