第1種の過誤と第2種の過誤をわかりやすく解説

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

# パラメータ
mu0 = 0       # H0の下での母平均
mu1 = 2       # H1の下での母平均
sigma = 1     # 標準偏差
n = 25
se = sigma / np.sqrt(n)  # 標準誤差
alpha = 0.05

# 臨界値
z_crit = stats.norm.ppf(1 - alpha)
x_crit = mu0 + z_crit * se

# x軸の範囲
x = np.linspace(mu0 - 4 * se, mu1 + 4 * se, 1000)

# H0とH1の下での標本平均の分布
pdf_h0 = stats.norm.pdf(x, mu0, se)
pdf_h1 = stats.norm.pdf(x, mu1, se)

fig, ax = plt.subplots(figsize=(12, 6))

# H0の分布
ax.plot(x, pdf_h0, 'b-', linewidth=2, label='$H_0$: $\\mu = 0$')
# H1の分布
ax.plot(x, pdf_h1, 'r-', linewidth=2, label='$H_1$: $\\mu = 2$')

# 第1種の過誤（α）: H0のもとで棄却域に入る部分
x_alpha = x[x >= x_crit]
ax.fill_between(x_alpha, stats.norm.pdf(x_alpha, mu0, se),
                alpha=0.3, color='blue', label=f'Type I Error ($\\alpha$ = {alpha})')

# 第2種の過誤（β）: H1のもとで棄却しない部分
x_beta = x[x <= x_crit]
ax.fill_between(x_beta, stats.norm.pdf(x_beta, mu1, se),
                alpha=0.3, color='red', label=f'Type II Error ($\\beta$ = {stats.norm.cdf(x_crit, mu1, se):.3f})')

# 臨界値の線
ax.axvline(x=x_crit, color='green', linewidth=2, linestyle='--',
           label=f'Critical value = {x_crit:.3f}')

ax.set_xlabel('Sample mean $\\bar{X}$', fontsize=12)
ax.set_ylabel('Density', fontsize=12)
ax.set_title('Type I and Type II Errors in Hypothesis Testing', fontsize=14)
ax.legend(fontsize=10, loc='upper right')
ax.grid(True, alpha=0.3)
plt.savefig("type1_type2_errors.png", dpi=150, bbox_inches="tight")
plt.show()

# β と検出力の計算
beta = stats.norm.cdf(x_crit, mu1, se)
power = 1 - beta
print(f"Alpha (Type I Error): {alpha}")
print(f"Beta (Type II Error): {beta:.4f}")
print(f"Power: {power:.4f}")

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

# パラメータ
mu0 = 0
mu1 = 1.5
sigma = 1
n = 25
se = sigma / np.sqrt(n)

# 有意水準を変化させる
alphas = np.linspace(0.001, 0.30, 200)
betas = []
powers = []

for a in alphas:
    z_crit = stats.norm.ppf(1 - a)
    x_crit = mu0 + z_crit * se
    b = stats.norm.cdf(x_crit, mu1, se)
    betas.append(b)
    powers.append(1 - b)

fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# αとβの関係
axes[0].plot(alphas, betas, 'b-', linewidth=2)
axes[0].set_xlabel('$\\alpha$ (Type I Error)', fontsize=12)
axes[0].set_ylabel('$\\beta$ (Type II Error)', fontsize=12)
axes[0].set_title('Trade-off between $\\alpha$ and $\\beta$', fontsize=14)
axes[0].grid(True, alpha=0.3)

# αと検出力の関係
axes[1].plot(alphas, powers, 'r-', linewidth=2)
axes[1].set_xlabel('$\\alpha$ (Type I Error)', fontsize=12)
axes[1].set_ylabel('Power ($1 - \\beta$)', fontsize=12)
axes[1].set_title('Power as a Function of $\\alpha$', fontsize=14)
axes[1].grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig("alpha_beta_tradeoff.png", dpi=150, bbox_inches="tight")
plt.show()

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

# パラメータ
mu0 = 0
sigma = 1
alpha = 0.05
z_alpha = stats.norm.ppf(1 - alpha)

# 異なる効果量での検出力
effect_sizes = [0.2, 0.5, 0.8, 1.0, 1.5]
n_values = np.arange(5, 201)

fig, ax = plt.subplots(figsize=(12, 6))

for delta in effect_sizes:
    powers = []
    for n in n_values:
        se = sigma / np.sqrt(n)
        mu1 = mu0 + delta
        x_crit = mu0 + z_alpha * se
        power = 1 - stats.norm.cdf(x_crit, mu1, se)
        powers.append(power)

    ax.plot(n_values, powers, linewidth=2, label=f'Effect size = {delta}')

ax.axhline(y=0.8, color='gray', linestyle='--', alpha=0.7, label='Power = 0.8 threshold')
ax.set_xlabel('Sample size (n)', fontsize=12)
ax.set_ylabel('Power ($1 - \\beta$)', fontsize=12)
ax.set_title(f'Power vs Sample Size ($\\alpha$ = {alpha})', fontsize=14)
ax.legend(fontsize=10)
ax.grid(True, alpha=0.3)
ax.set_ylim(0, 1.05)
plt.savefig("power_vs_sample_size.png", dpi=150, bbox_inches="tight")
plt.show()

import numpy as np
from scipy import stats

np.random.seed(42)

# パラメータ
mu0 = 0
mu1 = 1.0
sigma = 1
n = 30
alpha = 0.05
n_simulations = 10000

z_crit = stats.norm.ppf(1 - alpha)

# 第1種の過誤のシミュレーション（H0が真の場合）
type1_errors = 0
for _ in range(n_simulations):
    sample = np.random.normal(mu0, sigma, n)
    z = (sample.mean() - mu0) / (sigma / np.sqrt(n))
    if z >= z_crit:
        type1_errors += 1

# 第2種の過誤のシミュレーション（H1が真の場合）
type2_errors = 0
for _ in range(n_simulations):
    sample = np.random.normal(mu1, sigma, n)
    z = (sample.mean() - mu0) / (sigma / np.sqrt(n))
    if z < z_crit:
        type2_errors += 1

print("=== Simulation Results ===")
print(f"Number of simulations: {n_simulations}")
print(f"Sample size: n = {n}")
print(f"Significance level: alpha = {alpha}")
print()
print(f"Type I Error Rate: {type1_errors/n_simulations:.4f} (theoretical: {alpha})")

# 理論的なβ
delta = (mu1 - mu0) * np.sqrt(n) / sigma
beta_theory = stats.norm.cdf(z_crit - delta)
print(f"Type II Error Rate: {type2_errors/n_simulations:.4f} (theoretical: {beta_theory:.4f})")
print(f"Power (empirical): {1 - type2_errors/n_simulations:.4f} (theoretical: {1 - beta_theory:.4f})")