One-Sample Z-Test — When and How to Use It

One-Sample Z-Test

The one-sample z-test tests whether a population mean equals a hypothesized value when the population standard deviation (σ) is known and the sample is large (n ≥ 30).

$z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$

Assumptions

1. Data is from a random sample
2. Population standard deviation σ is known
3. Either: population is normal, OR n ≥ 30 (CLT applies)
4. Observations are independent

Step-by-Step Procedure

import numpy as np
from scipy import stats
import matplotlib.pyplot as plt

# Scenario: IQ scores. Population σ = 15 (known).
# We sample 50 students from a school. Do they differ from national μ₀ = 100?

np.random.seed(42)
sigma = 15      # known population std dev
mu_0 = 100      # null hypothesis mean
alpha = 0.05
n = 50

sample = np.random.normal(105, sigma, n)   # true mean is 105
x_bar = sample.mean()

# Step 1: State hypotheses
print("H₀: μ = 100  (school same as national average)")
print("H₁: μ ≠ 100  (two-tailed)")
print()

# Step 2: Compute test statistic
z = (x_bar - mu_0) / (sigma / np.sqrt(n))
print(f"x̄ = {x_bar:.4f}")
print(f"z = ({x_bar:.4f} - {mu_0}) / ({sigma}/√{n}) = {z:.4f}")

# Step 3: Critical value and rejection region
z_crit = stats.norm.ppf(1 - alpha/2)  # two-tailed
print(f"\nα = {alpha}, critical value z* = ±{z_crit:.4f}")
print(f"Reject H₀ if |z| > {z_crit:.4f}")

# Step 4: P-value
p_value = 2 * stats.norm.sf(abs(z))
print(f"p-value = {p_value:.6f}")

# Step 5: Decision
print(f"\n|z| = {abs(z):.4f} {'>' if abs(z) > z_crit else '≤'} {z_crit:.4f}")
print(f"p = {p_value:.4f} {'<' if p_value < alpha else '≥'} α = {alpha}")
print(f"Decision: {'Reject H₀' if p_value < alpha else 'Fail to reject H₀'}")
if p_value < alpha:
    print("Conclusion: Significant evidence students differ from national average")

# Step 6: Confidence interval
se = sigma / np.sqrt(n)
ci_lower = x_bar - z_crit * se
ci_upper = x_bar + z_crit * se
print(f"\n95% CI: ({ci_lower:.4f}, {ci_upper:.4f})")
print(f"Since 100 is {'NOT ' if not (ci_lower <= 100 <= ci_upper) else ''}in CI → {'Reject' if not (ci_lower <= 100 <= ci_upper) else 'Fail to reject'} H₀")

Critical Region Visualization

x = np.linspace(-4, 4, 1000)
y = stats.norm.pdf(x)

fig, ax = plt.subplots(figsize=(10, 5))
ax.plot(x, y, 'b-', linewidth=2)

# Rejection regions (two-tailed, α=0.05)
ax.fill_between(x, y, where=x <= -z_crit, alpha=0.4, color='red', label=f'Reject H₀ (each α/2={alpha/2})')
ax.fill_between(x, y, where=x >= z_crit, alpha=0.4, color='red')
ax.fill_between(x, y, where=(-z_crit <= x) & (x <= z_crit), alpha=0.2, color='blue', label='Fail to reject H₀')

ax.axvline(z, color='green', linewidth=2, linestyle='--', label=f'Observed z = {z:.3f}')
ax.axvline(-z_crit, color='black', linewidth=1.5, linestyle=':')
ax.axvline(z_crit, color='black', linewidth=1.5, linestyle=':', label=f'z* = ±{z_crit:.3f}')

ax.set_title(f'One-Sample Z-Test (Two-Tailed, α={alpha})')
ax.set_xlabel('z')
ax.set_ylabel('Density')
ax.legend()
plt.tight_layout()
plt.savefig('z_test.png', dpi=150)
plt.show()

One-Tailed Tests

# Left-tailed: H₁: μ < μ₀
def z_test(sample, mu_0, sigma, alternative='two-sided'):
    n = len(sample)
    x_bar = sample.mean()
    z = (x_bar - mu_0) / (sigma / np.sqrt(n))
    
    if alternative == 'two-sided':
        p = 2 * stats.norm.sf(abs(z))
    elif alternative == 'less':
        p = stats.norm.cdf(z)
    elif alternative == 'greater':
        p = stats.norm.sf(z)
    
    return z, p

z_stat, p_two   = z_test(sample, 100, 15, 'two-sided')
z_stat, p_less  = z_test(sample, 100, 15, 'less')
z_stat, p_greater = z_test(sample, 100, 15, 'greater')

print(f"z = {z_stat:.4f}")
print(f"p (two-sided): {p_two:.6f}")
print(f"p (H₁: μ<100): {p_less:.6f}")
print(f"p (H₁: μ>100): {p_greater:.6f}")

When Z-test vs T-test

In practice: σ is almost never truly known → use the t-test.

Key Takeaways

1. Z-test requires known σ — rare in practice; use t-test otherwise
2. Test statistic z measures how many standard errors x̄ is from μ₀
3. |z| > 1.96 → reject H₀ at α = 0.05 (two-tailed) — memorize this
4. Confidence interval and hypothesis test are equivalent — CI excludes μ₀ iff p < α
5. Effect size (d = (x̄ − μ₀)/σ) should accompany the z-statistic