Z-Test for One Population Mean

Use this calculator to perform a hypothesis test for a single population mean when population standard deviation is known (rare but theoretically important).

When to Use This Test

  • One sample from a population
  • Population SD is known (σ)
  • Testing against a hypothesized value (μ₀)
  • Large samples or normally distributed population
  • Examples: Testing if assembly time matches standard, testing if pH meets specification

Note: In practice, use t-test instead (when σ is unknown), which is more common.

How to Use This Calculator

Step 1: Enter the hypothesized population mean (μ₀)

Step 2: Enter the known population standard deviation (σ)

Step 3: Enter the sample size (n)

Step 4: Enter the sample mean (x̄)

Step 5: Enter the level of significance (α, typically 0.05)

Step 6: Select tail type:

  • Left-tailed: Testing if mean is less than μ₀
  • Right-tailed: Testing if mean is greater than μ₀
  • Two-tailed: Testing if mean differs from μ₀

Step 7: Click “Calculate”

Interpret results:

  • If p-value < α: Reject H₀ (significant difference)
  • If p-value ≥ α: Fail to reject H₀ (no significant evidence)
Z test Calculator for mean
Population Mean ($\mu$)
Population Standard Deviation ($\sigma$)
Sample Size ($n$)
Sample Mean ($\overline{x}$)
Level of Significance ($\alpha$)
Tail : Left tailed Right tailed Two tailed
Results
Standard Error of Mean:
Test Statistics Z:
Z-critical value:
p-value:

Test Formula

$$Z = \frac{\overline{x} - \mu_0}{\sigma/\sqrt{n}}$$

Where:

  • $\overline{x}$ = sample mean
  • $\mu_0$ = hypothesized population mean
  • $\sigma$ = known population standard deviation
  • $n$ = sample size

Decision Rule

For two-tailed test (α = 0.05):

  • Reject H₀ if p-value < 0.05
  • Reject H₀ if |Z| > 1.96
  • Otherwise: Fail to reject H₀

Assumptions

  1. Population SD known (σ is known)
  2. Random sample collected
  3. Independent observations
  4. Large sample OR normal population (n ≥ 30 or data approximately normal)

Worked Example

Scenario: Testing assembly time. Industry standard = 10 minutes. Historical σ = 1.5 min. Sample: n=50, x̄=10.4 min.

Hypotheses:

  • H₀: μ = 10 (assembly time meets standard)
  • H₁: μ ≠ 10 (assembly time differs, two-tailed)

Calculation:

  • $SE = 1.5/\sqrt{50} = 0.212$
  • $Z = (10.4 - 10)/0.212 = 1.887$
  • For α=0.05, two-tailed: critical Z = ±1.96
  • p-value = 0.0593

Decision: p-value (0.0593) > α (0.05), fail to reject H₀. No significant evidence that assembly time differs from 10 minutes.

Common Interpretation Mistakes

❌ WRONG: “We accept H₀” ✓ RIGHT: “We fail to reject H₀”

❌ WRONG: “p-value = 0.05 means 5% chance H₀ is true” ✓ RIGHT: “If H₀ is true, p-value = 0.05 means this result occurs 5% of the time”

❌ WRONG: “Non-significant = null hypothesis is true” ✓ RIGHT: “Non-significant = insufficient evidence to reject H₀”

⚠️ Important: In practice, use t-test for single mean (σ unknown) - almost always more appropriate.

Related: T-test for One Mean, Tutorial