One Sample Z Test Calculator
Calculate One-Sample Z-test statistic ($Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$), margin of error, $p$-value, and 95% confidence interval.
Calculate One-Sample Z-Test Benchmark
Enter sample mean (x̄), hypothesized mean (μ0), known population SD (σ), and sample size (n).
Calculation Results
Calculated using One-Sample Z-test equations: Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}, \quad \text{ME} = 1.96 \cdot \frac{\sigma}{\sqrt{n}}
Quick Summary
The One Sample Z Test Calculator evaluates $Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$, margin of error, $p$-value, and 95% confidence intervals.
Formula Explanation
Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}
\text{95\% CI} = \bar{x} \pm 1.96 \cdot \left(\frac{\sigma}{\sqrt{n}}\right)How It Works
The One-Sample Z-test checks if a sample mean $\bar{x}$ drawn from a population with known SD $\sigma$ differs from hypothesized mean $\mu_0$. The One Sample Z Test Calculator computes standard error, $Z$-score, margin of error, and confidence bounds.
Step-by-Step Worked Example
Practical Problem: Sample mean $\bar{x} = 105$, benchmark $\mu_0 = 100$, population SD $\sigma = 15$, sample size $n = 50$. Test for significance ($\alpha = 0.05$, 2-tailed).
- Step 1: Calculate Standard Error ($\text{SE}$): $\text{SE} = 15 / \sqrt{50} = 15 / 7.0711 = \mathbf{2.1213}$.
- Step 2: Calculate difference from benchmark: $105 - 100 = \mathbf{+5.0000}$.
- Step 3: Calculate Z-statistic: $Z = +5.0 / 2.1213 = \mathbf{+2.3570}$.
- Step 4: Calculate two-tailed p-value: $p = 2 \times (1 - \Phi(2.3570)) = 2 \times 0.0092 = \mathbf{0.0184}$.
- Step 5: Statistical Decision: Since $p = 0.0184 < 0.05$, reject $H_0$ (sample mean is significantly higher than benchmark).
Real-World Calculation Examples
Scenario 1: Benchmark Test (n = 50, x̄ = 105, σ = 15)
Parameters: x̄ = 105, μ0 = 100, n = 50
Result: Z = +2.3570, p = 0.0184 (Reject H0 at 5%).
Scenario 2: Factory Machine Calibration (n = 64)
Parameters: x̄ = 10.1cm, μ0 = 10.0cm, σ = 0.4cm
Result: Z = +2.0000, p = 0.0455.
Scenario 3: Standardized Academic Test Evaluation
Parameters: x̄ = 520, μ0 = 500, σ = 100, n = 100
Result: Z = +2.0000, p = 0.0455.
Scenario 4: Call Center Resolution Time Audit
Parameters: Target μ0 = 5.0 mins, n = 200 calls
Result: One-sample Z-test evaluation.
Key Benefits of Using This Calculator
Margin of Error Output
Displays exact margin of error $\text{ME} = 1.96 \cdot \text{SE}$.
Automated Hypothesis Decision
Reports explicit hypothesis decision (Reject $H_0$ vs Fail to Reject $H_0$).
95% Confidence Interval
Outputs 95% confidence interval for true mean $\mu$.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is a One-Sample Z-test?
A One-Sample Z-test evaluates whether a sample mean $\bar{x}$ differs significantly from a known population mean $\mu_0$.
Why is population SD σ required?
Z-tests require known population standard deviation $\sigma$; if $\sigma$ is unknown, use a 1-sample t-test.
What is null hypothesis ($H_0$)?
$H_0: \mu = \mu_0$ (sample mean equals benchmark).
How do I calculate a 1-sample Z-test in Excel?
Use formula =Z.TEST(array, mu0, sigma).
What is 95% confidence interval formula?
$\bar{x} \pm 1.96 \cdot \left(\frac{\sigma}{\sqrt{n}}\right)$.
What is critical Z-value for 5% significance level?
Two-tailed critical $Z = \pm 1.96$; one-tailed critical $Z = 1.645$.
What happens if sample size n increases?
Larger $n$ reduces standard error ($\sigma / \sqrt{n}$), increasing $Z$-statistic magnitude and statistical power.
What is margin of error?
Margin of Error is half the width of the 95% confidence interval ($\text{ME} = 1.96 \cdot \text{SE}$).
Can Z-test be used if population is not normal?
Yes — by Central Limit Theorem, if $n \ge 30$, sample mean $\bar{x}$ is approximately normal regardless of population shape.
What is p-value interpretation?
The $p$-value is the probability of observing a sample mean as extreme as $\bar{x}$ assuming $H_0$ is true.