Z-Test Examples

Statistics · Axiom Academy

Master hypothesis testing for population means with known standard deviation Example 1: Quality Control - Bottle Filling Machine A bottling company claims their machine fills bottles to exactly 500 mL. The population standard deviation is known to be 8 mL. A quality inspector randomly samples 36 bottles and finds a mean volume of 497.5 mL. Test at the 0.05 significance level whether the machine is filling bottles correctly. Excellent work! You've completed this z-test example. Here's what we learned: Two-tailed tests: When testing if a parameter equals a specific value (not greater or less than), we use a two-tailed test and check both tails of the distribution. Test statistic formula: The z-test statistic is z = (x̄ - μ₀) / (σ / √n) , which measures how many standard errors the sample mean is from the hypothesized population mean. Decision rules: We reject H₀ if |z| > critical value OR if p-value < α. Both methods lead to the same conclusion. Practical interpretation: A statistically significant result means the machine is not filling bottles to the claimed 500 mL specification, suggesting calibration is needed. One-tailed vs. two-tailed: Use one-tailed tests when you have a directional hypothesis (greater than or less than), and two-tailed tests when checking for any difference from the claimed value.

This is the written version of the interactive lesson above. See the full Statistics course.