Introduction to Z-Tests

A z-test is a parametric statistical test used to test hypotheses about population means (when population variance is known) or proportions. Z-tests are based on the standard normal distribution and are commonly used when:

  • Sample size is large (n ≥ 30)
  • Population standard deviation is known
  • You’re testing a population proportion
  • The population is approximately normally distributed

Z-tests are one of the most fundamental tools in hypothesis testing and serve as the foundation for understanding more complex statistical tests.

When to Use Z-Tests

Conditions for Using Z-Tests

Z-tests are appropriate when:

  1. Large Sample Size: n ≥ 30 (Central Limit Theorem applies)
  2. Known Population Variance: σ is known (rare in practice)
  3. Normal Distribution: Population is normally distributed or sample is large enough
  4. Independence: Observations are independent
  5. Single or Two Samples: Not for multiple comparisons (use ANOVA instead)

Comparison with T-Tests

Feature Z-Test T-Test
Sample Size Large (n ≥ 30) Small to large
Population σ Known Unknown
Distribution Standard normal Student’s t
Degrees of freedom None n - 1
Robustness Less robust for small samples More robust

Section 1: One-Sample Z-Test

Purpose and Hypothesis

The one-sample z-test tests whether a sample mean differs significantly from a hypothesized population mean.

Hypotheses:

  • H₀: μ = μ₀ (Null hypothesis: population mean equals hypothesized value)
  • H₁: μ ≠ μ₀ (Two-tailed alternative: population mean differs)
  • H₁: μ > μ₀ (Right-tailed alternative)
  • H₁: μ < μ₀ (Left-tailed alternative)

Formula

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

Where:

  • $\overline{x}$ = Sample mean
  • μ₀ = Hypothesized population mean
  • σ = Population standard deviation
  • n = Sample size

Step-by-Step Procedure

Step 1: State Hypotheses

  • Define null and alternative hypotheses
  • Specify significance level (α, typically 0.05)

Step 2: Calculate Test Statistic

  • Compute sample mean: $\overline{x}$
  • Apply z-test formula

Step 3: Find P-value

  • For two-tailed test: P-value = 2 × P(Z > |z|)
  • For right-tailed test: P-value = P(Z > z)
  • For left-tailed test: P-value = P(Z < z)

Step 4: Decision

  • If P-value ≤ α: Reject H₀
  • If P-value > α: Fail to reject H₀

Example: One-Sample Z-Test for Mean

Problem: A cereal manufacturer claims their boxes contain 368g on average. A random sample of 36 boxes has mean weight of 364.5g. The population standard deviation is known to be 5g. Test at α = 0.05 whether the mean weight differs from the claimed value.

Solution:

Given:

  • n = 36, $\overline{x}$ = 364.5g, μ₀ = 368g, σ = 5g, α = 0.05

Calculate z-statistic: $$z = \frac{364.5 - 368}{5/\sqrt{36}} = \frac{-3.5}{0.833} = -4.20$$

For two-tailed test with α = 0.05:

  • Critical values: z = ±1.96
  • Since |z| = 4.20 > 1.96, reject H₀
  • P-value ≈ 0.00003 < 0.05

Conclusion: There is strong evidence that the mean weight differs from 368g.

Section 2: Two-Sample Z-Test for Means

Purpose and Hypothesis

The two-sample z-test compares means between two independent populations.

Hypotheses:

  • H₀: μ₁ = μ₂ (Population means are equal)
  • H₁: μ₁ ≠ μ₂ (Population means differ)

Assumptions

  1. Both populations normally distributed (or large samples)
  2. Both population standard deviations known
  3. Samples are independent
  4. Equal sample sizes preferred (but not required)

Formula

$$z = \frac{(\overline{x}_1 - \overline{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$$

Where:

  • $\overline{x}_1$, $\overline{x}_2$ = Sample means
  • σ₁, σ₂ = Population standard deviations
  • n₁, n₂ = Sample sizes
  • Under H₀: (μ₁ - μ₂) = 0

Simplified under H₀: $$z = \frac{\overline{x}_1 - \overline{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$$

Step-by-Step Procedure

Step 1: State Hypotheses

  • Null: μ₁ = μ₂
  • Alternative: μ₁ ≠ μ₂ (or one-tailed)
  • Significance level: α = 0.05

Step 2: Check Assumptions

  • Verify independence and normality
  • Both σ’s are known

Step 3: Calculate Test Statistic

  • Compute both sample means
  • Apply two-sample z formula

Step 4: Find P-value and Conclude

  • Compare to critical value or p-value
  • Make statistical decision

Example: Two-Sample Z-Test

Problem: Brand A energy drink claims to have more caffeine than Brand B. A sample of 50 Brand A drinks has mean caffeine 85mg (σ = 8mg). A sample of 50 Brand B drinks has mean caffeine 80mg (σ = 7mg). Test at α = 0.05 whether Brand A has more caffeine (right-tailed test).

Solution:

Given:

  • n₁ = 50, $\overline{x}_1$ = 85mg, σ₁ = 8mg
  • n₂ = 50, $\overline{x}_2$ = 80mg, σ₂ = 7mg
  • α = 0.05 (right-tailed)

Standard error: $$SE = \sqrt{\frac{64}{50} + \frac{49}{50}} = \sqrt{1.28 + 0.98} = \sqrt{2.26} = 1.503$$

Z-statistic: $$z = \frac{85 - 80}{1.503} = \frac{5}{1.503} = 3.33$$

Critical value (right-tailed, α = 0.05): z = 1.645

Since z = 3.33 > 1.645, reject H₀

Conclusion: Brand A has significantly more caffeine than Brand B (p < 0.001).

Section 3: Z-Test for Proportions

One-Sample Proportion Test

Purpose

Tests whether a population proportion differs from a hypothesized value.

Hypotheses:

  • H₀: p = p₀
  • H₁: p ≠ p₀ (or one-tailed)

Requirements

  • np₀ ≥ 5
  • n(1 - p₀) ≥ 5
  • Random sample

Formula

$$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$$

Where:

  • $\hat{p}$ = Sample proportion (x/n)
  • p₀ = Hypothesized population proportion
  • n = Sample size

Example: One-Sample Proportion Test

Problem: A politician claims 50% of voters support their policy. In a sample of 200 voters, 115 support it. Test at α = 0.05 whether the true proportion differs from 50%.

Solution:

Given:

  • n = 200, x = 115, $\hat{p}$ = 0.575, p₀ = 0.50, α = 0.05

Check requirements:

  • np₀ = 200 × 0.50 = 100 ≥ 5 ✓
  • n(1-p₀) = 200 × 0.50 = 100 ≥ 5 ✓

Standard error: $$SE = \sqrt{\frac{0.50 × 0.50}{200}} = \sqrt{0.00125} = 0.0354$$

Z-statistic: $$z = \frac{0.575 - 0.50}{0.0354} = \frac{0.075}{0.0354} = 2.12$$

Critical values (two-tailed, α = 0.05): z = ±1.96

Since |z| = 2.12 > 1.96, reject H₀

Conclusion: The proportion supporting the policy is significantly different from 50% (p = 0.034).

Two-Sample Proportion Test

Purpose

Compares proportions between two independent populations.

Hypotheses:

  • H₀: p₁ = p₂
  • H₁: p₁ ≠ p₂

Formula

$$z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$$

Where: $$\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$$ (Pooled proportion)

Example: Two-Sample Proportion Test

Problem: Test whether treatment and control groups have different success rates.

  • Treatment: 80 successes out of 150
  • Control: 60 successes out of 150
  • α = 0.05

Solution:

Given:

  • $\hat{p}_1$ = 80/150 = 0.533, $\hat{p}_2$ = 60/150 = 0.400

Pooled proportion: $$\hat{p} = \frac{80 + 60}{150 + 150} = \frac{140}{300} = 0.467$$

Standard error: $$SE = \sqrt{0.467 × 0.533 × \left(\frac{1}{150} + \frac{1}{150}\right)} = \sqrt{0.249 × 0.0133} = 0.0577$$

Z-statistic: $$z = \frac{0.533 - 0.400}{0.0577} = \frac{0.133}{0.0577} = 2.30$$

Critical values (two-tailed): z = ±1.96

Since |z| = 2.30 > 1.96, reject H₀

Conclusion: The treatment group has a significantly higher success rate (p = 0.021).

Section 4: Critical Values and Decision Rules

Understanding Z Critical Values

Critical value is the threshold that separates rejection and acceptance regions of the null hypothesis.

Common Critical Values

Significance Level One-Tailed Two-Tailed
α = 0.10 ±1.28 ±1.645
α = 0.05 ±1.645 ±1.96
α = 0.01 ±2.33 ±2.576

Finding Critical Values

For One-Tailed Tests:

  • Right-tailed: Find z such that P(Z > z) = α
  • Left-tailed: Find z such that P(Z < z) = α

For Two-Tailed Tests:

  • Find z such that P(Z > z) = α/2 (both tails)
  • Critical values are ±z

Using Z-Score Tables

Standard normal tables provide cumulative probabilities P(Z ≤ z). To find critical values:

Example: Find critical value for α = 0.05 (two-tailed)

  • α/2 = 0.025
  • Find where P(Z ≤ z) = 1 - 0.025 = 0.975
  • This corresponds to z ≈ 1.96

Section 5: P-Values and Interpretation

What is a P-Value?

The p-value is the probability of observing test results as extreme or more extreme than what was actually observed, assuming the null hypothesis is true.

Interpretation:

  • P-value ≤ α: Reject H₀ (Results are statistically significant)
  • P-value > α: Fail to reject H₀ (Results are not statistically significant)

Calculating P-Values for Z-Tests

Two-Tailed Test: $$\text{P-value} = 2 × P(Z > |z_{calculated}|)$$

Right-Tailed Test: $$\text{P-value} = P(Z > z_{calculated})$$

Left-Tailed Test: $$\text{P-value} = P(Z < z_{calculated})$$

Example P-Value Calculation

If z = 2.5 (two-tailed test):

  • P(Z > 2.5) ≈ 0.00621
  • P-value = 2 × 0.00621 = 0.01242
  • At α = 0.05: Reject H₀

Common Misconceptions

❌ Incorrect: P-value is the probability that H₀ is true ✓ Correct: P-value is the probability of data given H₀ is true

❌ Incorrect: P-value = 0.08 means “almost significant” ✓ Correct: Use the predetermined α level (e.g., 0.05) consistently

Section 6: Effect Size for Z-Tests

Why Report Effect Size?

Statistical significance (small p-value) doesn’t indicate practical importance. Effect size measures the magnitude of the difference.

Cohen’s d

For comparing means, Cohen’s d quantifies the standardized difference:

$$d = \frac{\overline{x} - \mu_0}{\sigma}$$

Interpretation (Cohen’s Guidelines):

  • |d| ≈ 0.2: Small effect
  • |d| ≈ 0.5: Medium effect
  • |d| ≈ 0.8: Large effect

Example: Effect Size

From earlier example (cereal weights): $$d = \frac{364.5 - 368}{5} = \frac{-3.5}{5} = -0.70$$

This is a medium to large effect size, indicating practical significance beyond statistical significance.

Section 7: Assumptions and Conditions

Checking Assumptions

Before conducting a z-test, verify:

  1. Independence: Observations are independent

    • Random sampling or random assignment
    • Sample ≤ 10% of population (for finite populations)

  2. Normality: Population is normally distributed OR sample is large (n ≥ 30)

    • Check with Q-Q plot or Shapiro-Wilk test
    • Central Limit Theorem applies for large samples

  3. Known Variance: Population standard deviation is known

    • Rarely true in practice; use t-test if unknown

What if Assumptions are Violated?

  • Non-normal data + small sample: Use t-test or non-parametric alternative (Mann-Whitney U)
  • Unknown variance: Use t-test
  • Non-independent observations: Use paired t-test or other designs

Section 8: Practical Applications

Business: Quality Control

Scenario: A manufacturer wants to verify if a production line produces parts with correct dimensions (120mm target).

  • Sample: 64 parts, mean = 119.8mm, σ = 1.2mm
  • H₀: μ = 120
  • H₁: μ ≠ 120
  • Conclusion: Guide production adjustments

Medicine: Clinical Trials

Scenario: Testing if a new medication reduces blood pressure differently than placebo.

  • Treatment group: n₁ = 100, mean BP reduction = 12mmHg
  • Placebo group: n₂ = 100, mean BP reduction = 8mmHg
  • Conclusion: Determine drug efficacy

Marketing: Customer Satisfaction

Scenario: Testing if customer satisfaction improved after service changes.

  • Before: 65% satisfied
  • After: 72% satisfied (n = 500)
  • Conclusion: Assess impact of changes

Section 9: Common Mistakes and Pitfalls

Mistake 1: P-Hacking

Problem: Running multiple tests and reporting only significant results Solution: Pre-specify hypotheses and significance level before testing

Mistake 2: Ignoring Effect Size

Problem: Focusing only on p-value without considering practical significance Solution: Always report effect size (Cohen’s d) alongside test results

Mistake 3: Using Z-Test with Unknown Variance

Problem: Using z-test when population σ is unknown Solution: Use t-test for unknown variance (more appropriate)

Mistake 4: Violating Independence Assumption

Problem: Using z-test on paired or dependent data Solution: Use paired t-test for related samples

Mistake 5: Misinterpreting Non-Significance

Problem: Concluding “no difference exists” from p > 0.05 Solution: State “insufficient evidence” and consider sample size

Interactive Z-Test Calculator

[Calculator would be embedded here with examples for:]

  • One-sample z-test for means
  • Two-sample z-test for means
  • One-sample proportion test
  • Two-sample proportion test
  • Critical value finder
  • P-value calculator

Summary Table: Z-Test Selection

Scenario Test Type Formula Key Sample Size
One mean vs. target One-sample z z = (x̄ - μ)/SE n ≥ 30
Two means comparison Two-sample z z = (x̄₁ - x̄₂)/SE n₁, n₂ ≥ 30
One proportion vs. target One-prop z z = (p̂ - p₀)/SE np₀ ≥ 5
Two proportions comparison Two-prop z z = (p̂₁ - p̂₂)/SE Both ≥ 5

Key Formulas Cheat Sheet

One-Sample Z-Test

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

Two-Sample Z-Test

$$z = \frac{\overline{x}_1 - \overline{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$$

One-Sample Proportion Test

$$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$$

Two-Sample Proportion Test

$$z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$$

Standard Error (Two Samples)

$$SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$$

Cohen’s d

$$d = \frac{\overline{x} - \mu_0}{\sigma}$$

Alternative Comparison Tests:

Foundational Concepts:

Related Distribution Concepts:

Next Steps

After mastering z-tests:

  1. T-Tests: Learn when population variance is unknown
  2. Effect Sizes & Power Analysis: Understand practical significance
  3. ANOVA: Compare more than two groups
  4. Non-Parametric Tests: Alternatives when assumptions are violated

References

  1. Anderson, D.R., Sweeney, D.J., & Williams, T.A. (2018). Statistics for Business and Economics (14th ed.). Cengage Learning. - Comprehensive treatment of z-tests for means and proportions with real-world applications.

  2. NIST/SEMATECH. (2023). e-Handbook of Statistical Methods. Retrieved from https://www.itl.nist.gov/div898/handbook/ - Statistical methods for hypothesis testing including z-tests and normal distribution theory.