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Hypothesis Testing and Z-Tests

Sep 24, 2025

Overview

This lecture covers how to calculate and interpret a test statistic, specifically using the one-sample z-test, as part of hypothesis testing.

The Role of Alpha in Hypothesis Testing

  • Alpha (α) is the standard of evidence used to decide if a result is significant.
  • In most fields, including psychology, α is set at 0.05 and is not calculated by researchers.

Test Statistics and the Null Hypothesis

  • A test statistic summarizes how unlikely your sample is compared to what the null hypothesis predicts.
  • The null hypothesis states there is no effect; samples close to the population mean are expected if it is true.
  • A large difference between sample mean and population mean suggests a more "extreme" or unlikely result under the null.

The One-Sample Z-Test

  • The z-test determines if a sample mean differs significantly from a population mean under the null hypothesis.
  • Formula: z = (xÌ„ - μ) / SE, where xÌ„ is the sample mean, μ is the population mean, and SE is the standard error.

Calculating Standard Error

  • Standard error (SE) = population standard deviation (σ) divided by the square root of sample size (√n).
  • SE tells you the standard deviation of many possible sample means.

Worked Example: Neuro IQ Study

  • Sample mean (xÌ„) = 105.9; population mean (μ) = 100; standard deviation (σ) = 15; sample size (n) = 15.
  • SE = 15 / √15 = 3.87.
  • z = (105.9 - 100) / 3.87 = 1.53.
  • Do not confuse SE with standard deviation in the denominator; using standard deviation instead would calculate effect size, not the test statistic.

Key Terms & Definitions

  • Alpha (α) — the threshold for statistical significance, usually 0.05.
  • Null Hypothesis — the assumption that there is no effect or difference.
  • Test Statistic — a calculated number showing how extreme the sample data is relative to the null hypothesis.
  • One-Sample Z-Test — a test to compare a sample mean to a population mean.
  • Standard Error (SE) — the standard deviation of the sample mean's distribution, calculated as σ/√n.

Action Items / Next Steps

  • Prepare to learn how to make decisions based on the test statistic in the next lecture.

Full transcript