Two-Sample Z-Test

Statistics · Axiom Academy

Comparing population means when both standard deviations are known 1. When to Use the Two-Sample Z-Test The two-sample z-test is appropriate when comparing the means of two populations under specific conditions: You have samples from two independent populations Both population standard deviations ( ) are known Either the populations are normally distributed OR sample sizes are large (n ≥ 30) Samples are randomly selected and independent Example: Comparing average test scores from two different schools where historical data gives us the population standard deviations. When comparing two population means, we formulate hypotheses about the relationship between and : Two-tailed: H₁: μ₁ ≠ μ₂ (testing for any difference) Right-tailed: H₁: μ₁ > μ₂ (testing if population 1 is greater) Left-tailed: H₁: μ₁ < μ₂ (testing if population 1 is less) The two-sample z-statistic measures how many standard errors the difference in sample means is from zero (assuming H₀ is true): = sample mean from population 1 = sample mean from population 2 = known standard deviation of population 1 = known standard deviation of population 2 = sample size from population 1 = sample size from population 2 The denominator is the standard error of the difference between means, combining variability from both samples. 4. Confidence Intervals for Difference in Means Instead of just testing hypotheses, we can construct a confidence interval to estimate the true difference μ₁ - μ₂:

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