File:Interaction between sample size, effect size, and statistical power.svg
Summary
| Description |
English: Distributions of sample means under the null (θ=0) and alternative hypotheses are shown. The shaded red area represents Type I error (α), held constant at 0.05, while the shaded green area represents statistical power (1-β). As the sample size increases, the distributions narrow, leading to clearer separation between the hypotheses and higher power. Similarly, a larger effect size increases the distance between the distributions, resulting in greater power.
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ax.plot(x, alt_dist, label=rf'$H_1\; (\theta = {theta})```python import numpy as np import matplotlib.pyplot as plt from scipy.stats import norm
alpha = 0.05 # Significance level
xmin, xmax = -0.4, 0.6 ns = [10, 30, 100] thetas = [0.1, 0.3, 0.5] fig, axes = plt.subplots(len(ns), len(thetas), figsize=(12, 10))
for i, n in enumerate(ns): # Sample sizes for j, theta in enumerate(thetas): # Effect sizes
# Calculate parameters
sigma = 1 / np.sqrt(n)
z_alpha = norm.ppf(1 - alpha)
threshold = z_alpha * sigma
power = 1 - norm.cdf(z_alpha - theta / sigma)
# Generate distributions
x = np.linspace(xmin, xmax, 1000)
null_dist = norm.pdf(x, 0, sigma)
alt_dist = norm.pdf(x, theta, sigma)
# Plot on the corresponding subplot
ax = axes[i, j]
ax.plot(x, null_dist, label=rf'$H_0\; (\theta = 0) |
| Source |
Own work  |
| Author |
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ax.axvline(threshold, color='gray', linestyle='--', label='Threshold')
ax.fill_between(x, null_dist, where=(x > threshold), color='red', alpha=0.3, label='Significance (α)')
ax.fill_between(x, alt_dist, where=(x > threshold), color='green', alpha=0.3, label='Power (1-β)')
# Set subplot title and labels
ax.set_title(rf'$n={n}, \theta={theta}$ Power$={power:.2f}}})
ax.set_xlabel('Sample Mean (x̄)')
ax.set_yticks([])
ax.set_xlim(xmin, xmax)
# ax.set_ylabel('Probability Density')
if i == 0 and j == 0: ax.legend(fontsize='small')
plt.tight_layout() plt.savefig('power_analysis.svg') plt.show() ``` }} |date=2024-07-26 |source=Own work |author=Cosmia Nebula }}
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