Biostatistics Study NotesFrom concepts and derivations to interpretation

CHAPTER 04 · TOPIC 03

One-Sample t Test

Compare one population mean with a specified value using hypotheses, the t statistic, degrees of freedom, and a complete testing procedure.

On this page
  1. What Does “One Sample” Mean?
  2. State the Hypotheses
  3. Test Statistic
  4. Testing Procedure

The one-sample t-test examines whether the mean of a quantitative variable in one sample differs from a prespecified constant μ₀. For μ₀, national averages, product standards, reference values, research thresholds, etc. can be used.

What Does “One Sample” Mean?#

It means a group of representative samples obtained from one population, and does not mean that there is only one observed value. Measurements are quantitative variables such as scores, life spans, survival times, etc. for which averages can be reasonably calculated.

State the Hypotheses#

two-sidedH0:μ=μ0,H1:μ≠μ0H_0:\mu=\mu_0,\qquad H_1:\mu\ne\mu_0
right-tailedH0:μ≤μ0,H1:μ>μ0H_0:\mu\leq\mu_0,\qquad H_1:\mu>\mu_0
left-tailedH0:μ≥μ0,H1:μ<μ0H_0:\mu\geq\mu_0,\qquad H_1:\mu<\mu_0

Test Statistic#

σ knownZ=Xˉ−μ0σ/nZ=\frac{\bar X-\mu_0}{\sigma/\sqrt n}
σ unknownt=Xˉ−μ0s/nt=\frac{\bar X-\mu_0}{s/\sqrt n}
degree of freedomν=n−1\nu=n-1

Testing Procedure#

  1. Predetermine μ₀, H₀, H₁, direction, α
  2. Calculate n, X̄, s
  3. Find t=(X̄−μ₀)/(s/√n), df=n−1
  4. Find the p-value from the t-distribution and compare it with α to interpret the conclusion