Z-Test for Population Mean

Statistics · Axiom Academy

LESSON Z-Test for Population Mean Understanding hypothesis testing when population standard deviation is known The z-test for a population mean is appropriate under specific conditions. Let's visualize the decision process for choosing this test. The population standard deviation (σ) is known The sample size is large (n ≥ 30) by Central Limit Theorem The population is normally distributed (for small samples) Random sampling from the population is ensured The z-test statistic measures how many standard errors the sample mean is away from the hypothesized population mean. Let's see how each component contributes to the calculation. x̄ = Sample mean (observed value) μ₀ = Hypothesized population mean σ = Population standard deviation (known) σ/√n = Standard error of the mean 3. Steps for Conducting a Z-Test Follow these systematic steps to perform a z-test. Watch as each step builds upon the previous one. Step 1: State hypotheses (H₀ and H₁) Step 2: Choose significance level (α) Step 3: Calculate test statistic z Step 4: Find p-value or critical value Step 5: Make decision and conclusion 4. Finding P-Values Using Standard Normal Distribution The p-value represents the probability of observing a test statistic as extreme as (or more extreme than) what we calculated, assuming H₀ is true. We use the standard normal distribution to find it. Two-tailed test: p-value = 2 × P(Z > |z|) Right-tailed test: p-value = P(Z > z) Left-tailed test: p-value = P(Z < z)

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