The Z-Test › One-Sample Z-Test: Complete Worked Example
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Let's work through a complete example with every calculation shown explicitly.
The Problem
A coffee machine manufacturer claims their machines dispense exactly 200 mL per cup, with a known standard deviation of 5 mL (established through extensive quality testing). A consumer group tests 49 cups from a randomly selected machine and finds a mean of 197.5 mL. Is there evidence that this machine dispenses less than the claimed amount?
Step 1: State the Hypotheses
Since we're specifically asking whether the machine dispenses less than claimed, this is a one-sided test:
- (The machine dispenses the claimed amount)
- (The machine dispenses less than claimed)
Step 2: Identify the Known Values
- Sample mean: mL
- Hypothesized mean: mL
- Population standard deviation: mL
- Sample size:
- Significance level:
Step 3: Calculate the Standard Error
This tells us that sample means of 49 cups typically vary by about 0.714 mL from the true population mean.
Step 4: Calculate the Z-Statistic
The sample mean is 3.50 standard errors below the hypothesized value. This is quite extreme!
Step 5: Calculate the P-Value
For a left-tailed test:
Step 6: Make a Decision
Since , we reject the null hypothesis. There is strong evidence that the machine dispenses less than 200 mL per cup.
import numpy as np
from scipy import stats
# Given values
x_bar = 197.5 # Sample mean
mu_0 = 200 # Hypothesized mean
sigma = 5 # Population standard deviation
n = 49 # Sample size
alpha = 0.05 # Significance level
# Step 3: Standard error
se = sigma / np.sqrt(n)
# Step 4: Z-statistic
z = (x_bar - mu_0) / se
# Step 5: P-value (left-tailed)
p_value = stats.norm.cdf(z)
# Critical value for left-tailed test
z_critical = stats.norm.ppf(alpha)
print("One-Sample Z-Test: Coffee Machine")
print("=" * 50)
print(f"Sample mean: x̄ = {x_bar} mL")
print(f"Hypothesized mean: μ₀ = {mu_0} mL")
print(f"Population std: σ = {sigma} mL")
print(f"Sample size: n = {n}")
print()
print(f"Standard error: SE = {se:.4f} mL")
print(f"Z-statistic: z = {z:.3f}")
print(f"P-value (left-tailed): p = {p_value:.6f}")
print(f"Critical value: z_α = {z_critical:.3f}")
print()
print(f"Decision: {'Reject H₀' if p_value < alpha else 'Fail to reject H₀'}")
print()
print("Interpretation: There is strong evidence that the machine")
print("dispenses less than the claimed 200 mL per cup.")One-Sample Z-Test: Coffee Machine ================================================== Sample mean: x̄ = 197.5 mL Hypothesized mean: μ₀ = 200 mL Population std: σ = 5 mL Sample size: n = 49 Standard error: SE = 0.7143 mL Z-statistic: z = -3.500 P-value (left-tailed): p = 0.000233 Critical value: z_α = -1.645 Decision: Reject H₀ Interpretation: There is strong evidence that the machine dispenses less than the claimed 200 mL per cup.
Confidence Interval Interpretation
We can also construct a 95% confidence interval for the true mean:
# 95% CI for the population mean
z_crit_ci = stats.norm.ppf(0.975) # For two-sided CI
margin = z_crit_ci * se
ci_lower = x_bar - margin
ci_upper = x_bar + margin
print(f"95% Confidence Interval: [{ci_lower:.2f}, {ci_upper:.2f}] mL")
print()
print("The claimed value of 200 mL is outside this interval,")
print("which is consistent with rejecting the null hypothesis.")95% Confidence Interval: [196.10, 198.90] mL The claimed value of 200 mL is outside this interval, which is consistent with rejecting the null hypothesis.