One Proportion Test

Introduction: One Proportion Test

The One Proportion Test is a statistical method used to determine whether an observed proportion differs significantly from an expected or hypothesised value. It is widely applied in quality control, market research, and healthcare to test whether performance or outcomes meet a defined target or standard.

Background

The One Proportion Test belongs to inferential statistics, where conclusions about populations are drawn from sample data. By comparing observed proportions with expected ones, this test helps determine whether deviations are due to random chance or represent a statistically significant difference.
It is commonly used in Lean Six Sigma projects to verify whether process improvements lead to measurable quality gains.

Key Elements / Features

• Null Hypothesis (H₀) – There is no difference between the observed and expected proportions.
• Alternative Hypothesis (Hₐ) – There is a significant difference between the observed and expected proportions.
• p-value – The probability of observing the data if H0 is true. A small p-value (typically < 0.05) suggests evidence against the null hypothesis.

Formula (z-test for proportion):

\(
z = \frac{\hat{p} – p_0}{\sqrt{\frac{p_0 (1 – p_0)}{n}}}
\)

Where:

• \(
  \hat{p} = \text{observed sample proportion}
  \)
• \(
  p_0 = \text{expected or hypothesised proportion}
  \)
• \(
  n = \text{sample size}
  \)

Applications / Examples

• Manufacturing: Testing if the proportion of defect-free products meets a 95% quality target.
• Healthcare: Checking whether recovery rates from a treatment match expected outcomes.
• Market Research: Assessing whether customer preference proportions differ from predictions.

Example:
A factory claims 95% of its products are defect-free. In a sample of 400 products, 372 are defect-free   

(\(\hat{p} = 0.93\))

\(
z = \frac{0.93 – 0.95}{\sqrt{\frac{0.95(1 – 0.95)}{400}}} = -1.84
\)

At a significance level of 0.05, the result is not statistically significant, meaning the process still meets the 95% standard within random variation.

Relevance / Impact

The One Proportion Test provides a structured way to:

• Support decision-making: Determine whether observed outcomes meet targets.
• Ensure quality control: Maintain compliance with product or process standards.
• Manage risk: Detect deviations early before they escalate into systemic issues.

It is a key tool in Lean Six Sigma for validating process stability, customer satisfaction levels, or product conformance rates.

See also

• Hypothesis Testing
• Null Hypothesis
• p-Value
• Two Proportion Test