One Sample t-Test

Introduction: One Sample t-Test

The One Sample t-Test is a statistical method used to determine whether the mean of a sample significantly differs from a known or expected value. It is commonly applied in quality control, process optimisation, and Lean Six Sigma projects, where verifying whether a process meets a defined target is essential for maintaining standards and reducing variation.

Background

Developed within the field of inferential statistics, the One Sample t-Test is used when population parameters are unknown and the sample size is relatively small. The test assesses whether the observed mean (\(\bar{x}\)) differs from a specified target (\(\mu_0\)) by more than what would be expected due to random sampling variation.
It is particularly useful in manufacturing, healthcare, and service environments for validating process stability and improvement outcomes.

Key Elements / Features

• Null Hypothesis: The sample mean equals the target or expected value: \(
  H_0: \bar{x} = \mu_0
  \)
• Alternative Hypothesis: The sample mean differs from the target value: \(
  H_a: \bar{x} \neq \mu_0
  \)

Calculation: The test computes a t-value, representing the difference between the sample mean and the hypothesised mean, scaled by sample variability.

Formula (t-statistic):

\(
t = \frac{\bar{x} – \mu_0}{s / \sqrt{n}}
\)

Where:

• \(
  \bar{x} = \text{sample mean}
  \)
• \(
  \mu_0 = \text{hypothesised or target mean}
  \)
• \(
  s = \text{sample standard deviation}
  \)
• \(
  n = \text{sample size}
  \)

The resulting t-value is compared to critical values from the t-distribution (based on degrees of freedom = n−1) to determine statistical significance.

Applications / Examples

• Manufacturing: Testing whether the average length of a part equals the 5 cm design standard.
• Healthcare: Comparing average recovery times of patients against benchmark targets.
• Business and Services: Checking whether average satisfaction scores meet target levels.

Example:
A factory claims the average part length is 5.0 cm. A sample of 10 parts has:

• \(
  \bar{x} = 4.9 \text{ cm}
  \)
• \(
  s = 0.15 \text{ cm}
  \)

\(
t = \frac{4.9 – 5.0}{0.15 / \sqrt{10}} = -2.11
\)

At a significance level of 0.05 with 9 degrees of freedom, if ∣t∣>2.262, the difference is significant — suggesting a potential quality issue.

Relevance / Impact
The One Sample t-Test is a core statistical tool for:

• Supporting quality control: Verifies whether processes meet design specifications.
• Driving process optimisation: Highlights deviations requiring corrective action.
• Improving decision-making: Provides evidence-based insights instead of assumptions.

By using the One Sample t-Test, organisations ensure that process changes are validated with reliable statistical confidence.

See also

• Hypothesis Testing
• Null Hypothesis
• Two Sample t-Test
• p-Value