EZ Statistics

Two-Sample Z-Test

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Two-Sample Z-Test

Definition

Two-Sample Z-Test is a statistical test used to determine whether the means of two populations are significantly different from each other when both population standard deviations are known. It's particularly useful for large samples and when working with known population parameters.

Formula

Test Statistic:

z=(xˉ1−xˉ2)−(μ1−μ2)σ12n1+σ22n2z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}

Where:

  • xˉ1,xˉ2\bar x_1, \bar x_2 = sample means
  • μ1,μ2\mu_1, \mu_2 = population means
  • σ1,σ2\sigma_1, \sigma_2 = known population standard deviations
  • n1,n2n_1, n_2 = sample sizes

Confidence Interval for Mean Difference:

Two-sided: (xˉ1−xˉ2)±zα/2σ12n1+σ22n2\text{Two-sided: }(\bar{x}_1 - \bar{x}_2) \pm z_{\alpha/2} \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}One-sided: (xˉ1−xˉ2)±zασ12n1+σ22n2\text{One-sided: }(\bar{x}_1 - \bar{x}_2) \pm z_{\alpha} \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}
Where zα/2z_{\alpha/2} and zαz_{\alpha} are the critical values for the two-tailed and one-tailed tests, respectively.

Key Assumptions

Known Population Standard Deviations: Both σ₁ and σ₂ must be known
Independent Samples: The two samples must be independent
Random Sampling: Both samples should be randomly selected
Normality: Both populations should be normal or have large samples (n>30n > 30)

Practical Example

Comparing the efficiency of two production lines with known process variations:

Step 1: State the Data
  • Line 1: n1n_1 = 50, xˉ1\bar x_1 = 95.2 units/hour, σ1\sigma_1 = 4.0
  • Line 2: n2n_2 = 45, xˉ2\bar x_2 = 93.8 units/hour, σ2\sigma_2 = 3.8
Step 2: State Hypotheses
  • H0:μ1−μ2=0H_0: \mu_1 - \mu_2 = 0 (no difference)
  • Ha:μ1−μ2≠0H_a: \mu_1 - \mu_2 \neq 0 (there is a difference)
  • α=0.05\alpha = 0.05
Step 3: Calculate Test Statistic

Z-statistic:

z=(95.2−93.8)−04.0250+3.8245=1.73z = \frac{(95.2 - 93.8) - 0}{\sqrt{\frac{4.0^2}{50} + \frac{3.8^2}{45}}} = 1.73
Step 4: Calculate P-value

For two-tailed test:

p-value=2(1−Φ(∣z∣))=2(1−Φ(1.73))=0.084p\text{-value} = 2(1 - \Phi(|z|)) = 2(1 - \Phi(1.73)) = 0.084
Step 5: Calculate Confidence Interval
CI: (95.2−93.8)±1.964.0250+3.8245=[0.7,2.3]\text{CI: }(95.2 - 93.8) \pm 1.96 \sqrt{\frac{4.0^2}{50} + \frac{3.8^2}{45}} = [0.7, 2.3]
Step 6: Draw Conclusion

Critical value at 5% significance level:

zα/2=1.96z_{\alpha/2} = 1.96

Since ∣z∣<zα/2|z| < z_{\alpha/2} and p-value>0.05p\text{-value} > 0.05, we fail to reject H0H_0. There is no significant difference between the two production lines.

Effect Size

Cohen's d for two-sample z-test:

d=∣xˉ1−xˉ2∣σ12+σ222d = \frac{|\bar{x}_1 - \bar{x}_2|}{\sqrt{\frac{\sigma_1^2 + \sigma_2^2}{2}}}

Interpretation guidelines:

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

Power Analysis

Required sample size per group for equal sample sizes:

Two-sided: n=2(z1−α/2+z1−β)2(σ12+σ22)(μ1−μ2)2\text{Two-sided: } n = \frac{2(z_{1-\alpha/2} + z_{1-\beta})^2(\sigma_1^2 + \sigma_2^2)}{(\mu_1-\mu_2)^2}One-sided: n=2(z1−α+z1−β)2(σ12+σ22)(μ1−μ2)2\text{One-sided: } n = \frac{2(z_{1-\alpha} + z_{1-\beta})^2(\sigma_1^2 + \sigma_2^2)}{(\mu_1-\mu_2)^2}

Where:

  • α\alpha = significance level
  • β\beta = probability of Type II error
  • μ1−μ2\mu_1 - \mu_2 = minimum detectable difference

Decision Rules

Reject H0H_0 if:

  • Two-sided test: ∣z∣>zα/2|z| > z_{\alpha/2}
  • Left-tailed test: z<−zαz < -z_{\alpha}
  • Right-tailed test: z>zαz > z_{\alpha}
  • Or if p-value<αp\text{-value} < \alpha

Reporting Results

Standard format:

"A two-sample z-test was conducted to compare [variable] between [group 1] (M₁ = [mean₁], σ₁ = [std₁], n₁ = [n₁]) and [group 2] (M₂ = [mean₂], σ₂ = [std₂], n₂ = [n₂]). Results indicated [significant/no significant] difference between the groups, z = [z-value], p = [p-value], d = [Cohen's d]."

Code Examples

R
1library(tidyverse)
2
3set.seed(42)
4# Production Line 1 data (known σ₁ = 4.0)
5line1 <- tibble(
6  line = "Line 1",
7  units = rnorm(50, mean = 95.2, sd = 4.0)
8)
9
10# Production Line 2 data (known σ₂ = 3.8)
11line2 <- tibble(
12  line = "Line 2",
13  units = rnorm(45, mean = 93.8, sd = 3.8)
14)
15
16# Combine data
17production_data <- bind_rows(line1, line2)
18
19# Summarize the data
20summary_stats <- production_data |>
21  group_by(line) |>
22  summarise(
23    n = n(),
24    mean = mean(units),
25   ".groups" = "drop"
26  ) |>
27  mutate(known_sd = if_else(line == "Line 1", line1_pop_sd, line2_pop_sd))
28
29# Perform two-sample Z-test
30line1_stats <- summary_stats |> filter(line == "Line 1")
31line2_stats <- summary_stats |> filter(line == "Line 2")
32
33# Calculate z-statistic
34z_stat <- (line1_stats$mean - line2_stats$mean) / sqrt((line1_stats$known_sd^2 / line1_stats$n) + (line2_stats$known_sd^2 / line2_stats$n))
35print(str_glue("Z-statistic: {round(z_stat, 3)}"))
36
37# 95% confidence interval
38alpha <- 0.05
39z_alpha <- qnorm(1 - alpha/2)
40mean_diff <- line1_stats$mean - line2_stats$mean
41margin_of_error <- z_alpha * sqrt((line1_stats$known_sd^2 / line1_stats$n) + (line2_stats$known_sd^2 / line2_stats$n))
42ci_lower <- mean_diff - margin_of_error
43ci_upper <- mean_diff + margin_of_error
44print(str_glue("95% CI: [{round(ci_lower, 2)}, {round(ci_upper, 2)}]")
45
46# Calculate p-value (two-sided test)
47p_value <- 2 * (1 - pnorm(abs(z_stat)))
48print(str_glue("P-value: {round(p_value, 4)}")
49
50# Calculate effect size (Cohen's d)
51pooled_sd <- sqrt((4.0^2 + 3.8^2) / 2)
52cohens_d <- abs(line1_stats$mean - line2_stats$mean) / pooled_sd
53print(str_glue("Effect size (Cohen's d): {round(cohens_d, 3)}"
54
55
56# Visualization
57ggplot(production_data, aes(x = line, y = units, fill = line)) +
58  geom_boxplot(alpha = 0.5) +
59  geom_jitter(width = 0.2, alpha = 0.5) +
60  theme_minimal() +
61  labs(
62    title = "Production Output by Line",
63    y = "Units per Hour",
64    x = "Production Line"
65  )
Python
1import numpy as np
2import scipy.stats as stats
3import pandas as pd
4import matplotlib.pyplot as plt
5import seaborn as sns
6
7# Set random seed for reproducibility
8np.random.seed(42)
9
10# Generate sample data
11# Production Line 1 (known σ₁ = 4.0)
12line1_data = np.random.normal(95.2, 40, 50)
13
14# Production Line 2 (known σ₂ = 3.8)
15line2_data = np.random.normal(93.8, 3.8, 45)
16
17# Calculate sample means
18sample_mean1 = np.mean(line1_data)
19sample_mean2 = np.mean(line2_data)
20
21# Calculate z-statistic
22z_numerator = (sample_mean1 - sample_mean2)
23z_denominator = np.sqrt((4.0**2/50) + (3.8**2/45))
24z_stat = z_numerator / z_denominator
25print(f"Z-statistic: {z_stat:.2f}")
26
27# Calculate p-value (two-sided test)
28p_value = 2 * (1 - stats.norm.cdf(abs(z_stat)))
29print(f"P-value: {p_value:.4f}")
30
31# Calculate 95% Confidence Interval
32alpha = 0.05
33z_critical = stats.norm.ppf(1 - alpha/2)
34margin_of_error = z_critical * z_denominator
35ci_lower = z_numerator - margin_of_error
36ci_upper = z_numerator + margin_of_error
37print(f"95% Confidence Interval for mean difference: ({ci_lower:.2f}, {ci_upper:.2f})")
38
39# Calculate effect size (Cohen's d)
40pooled_sd = np.sqrt((4.0**2 + 3.8**2) / 2)
41cohens_d = abs(sample_mean1 - sample_mean2) / pooled_sd
42print(f"Cohen's d: {cohens_d:.2f}")
43
44# Create DataFrame for plotting
45df = pd.DataFrame({
46    'Production Line': ['Line 1']*50 + ['Line 2']*45,
47    'Units': np.concatenate([line1_data, line2_data])
48})
49
50# Create visualization
51plt.figure(figsize=(12, 5))
52
53# Subplot 1: Boxplot
54plt.subplot(1, 2, 1)
55sns.boxplot(data=df, x='Production Line', y='Units')
56plt.title('Units per Hour by Production Line')
57
58# Subplot 2: Distribution
59plt.subplot(1, 2, 2)
60sns.histplot(data=df, x='Units', hue='Production Line',
61            element="step", stat="density")
62plt.title('Distribution of Units per Hour')
63
64plt.tight_layout()
65plt.show()

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