Broad overview

• One-sample tests are used to compare data from one sample to a known or expected population value
• These tests allow researchers to make inferences about populations using sample data
• The focus is on one variable measured in one sample

Key Terms

• Population: the full group of interest
• Sample: the subset of the population that is measured
• Variable: the characteristic measured in the study
• Mean: the average of a numeric variable
• Proportion: the percentage in each category of a categorical variable

Hypotheses in Statistical Testing

• Null hypothesis (H₀): assumes no difference or no effect
• Alternative hypothesis (H₁): assumes a difference or effect exists
• Statistical tests evaluate evidence against the null hypothesis

Probability and the p-value

• The p-value represents the probability of obtaining a result as extreme as the observed one if H₀ is true
• Smaller p-values indicate stronger evidence against H₀

Decision Rule

• A significance level (α) is set before analysis, commonly 0.05
• If p ≤ α, reject H₀
• If p > α, fail to reject H₀
• The null hypothesis is never “accepted”

One-Sample Tests

• All one-sample tests involve:
  • One sample
  • One variable
  • A comparison to a reference value or expected distribution

Selecting the Appropriate One-Sample Test

Variable Type	Comparison	Statistical Test
Numeric	Sample mean vs known mean, population SD known	One-sample z-test
Numeric	Sample mean vs known mean, population SD unknown	One-sample t-test
Categorical	Observed proportions vs expected proportions	Chi-square goodness-of-fit test

One-Sample z-Test

Purpose

• Tests whether a sample mean differs from a known population mean when population SD is known

Assumptions

• Variable is numeric
• Observations are independent
• Population is approximately normally distributed or sample size is large

One-Sample t-Test

Purpose

• Tests whether a sample mean differs from a known comparison mean when population SD is unknown

Assumptions

• Variable is numeric
• Observations are independent
• Distribution is approximately normal or sample size is large

Chi-Square Goodness-of-Fit Test

Purpose

• Tests whether observed category proportions differ from expected proportions

Assumptions

• Variable is categorical
• Observations are independent
• Expected frequency in each category is at least 5

The Normal Distribution and z-Scores

• A z-score indicates how many standard deviations a value is from the mean
• z-scores allow comparison across different scales
• The normal distribution underpins many statistical tests

Reporting Statistical Results

• z-test: z value and p-value
• t-test: t value, degrees of freedom, and p-value
• Chi-square test: χ² value, degrees of freedom, sample size, and p-value