CI = x̄ ± t* × (s/√n)

Where:
- x̄ = sample mean
- t* = critical value from t-distribution (depends on confidence level and df)
- s = sample standard deviation
- n = sample size
- df = degrees of freedom = n - 1

from scipy.stats import t
import numpy as np

def confidence_interval(data, confidence=0.95):
    n = len(data)
    mean = data.mean()
    std = data.std(ddof=1)  # sample standard deviation
    
    # Critical value from t-distribution
    alpha = 1 - confidence
    t_crit = t.ppf(1 - alpha/2, df=n-1)
    
    # Margin of error
    margin = t_crit * std / np.sqrt(n)
    
    return mean - margin, mean + margin

# Calculate CI for sleep hours
sleep_data = df['suenio_horas']
lower, upper = confidence_interval(sleep_data, confidence=0.95)

print(f"95% CI for mean sleep hours: [{lower:.2f}, {upper:.2f}]")
# Result: [6.36, 6.75]

# Confidence intervals at different levels
stress_data = df['estres_score']

for confidence_level in [0.90, 0.95, 0.99]:
    lower, upper = confidence_interval(stress_data, confidence=confidence_level)
    width = upper - lower
    print(f"{int(confidence_level*100)}% CI: [{lower:.2f}, {upper:.2f}] (width={width:.2f})")

90% CI: [18.00, 19.71] (width=1.71)
95% CI: [17.83, 19.87] (width=2.04)
99% CI: [17.50, 20.20] (width=2.70)

import numpy as np

np.random.seed(123)

for n_sample in [30, 60, 150]:
    sample = df['suenio_horas'].sample(n_sample, replace=False)
    lower, upper = confidence_interval(sample, confidence=0.95)
    width = upper - lower
    print(f"n={n_sample}: 95% CI width = {width:.3f}")

n=30:  width = 0.863
n=60:  width = 0.598
n=150: width = 0.389

H₀: μ = 7 (students get adequate sleep)
H₁: μ < 7 (students are sleep deprived)

from scipy.stats import ttest_1samp

# Sample data
sleep_hours = df['suenio_horas']
n = len(sleep_hours)
mean_sleep = sleep_hours.mean()  # 6.55
std_sleep = sleep_hours.std(ddof=1)  # 1.21

# Null hypothesis value
mu_0 = 7

# Calculate t-statistic
t_stat = (mean_sleep - mu_0) / (std_sleep / np.sqrt(n))
print(f"t-statistic: {t_stat:.3f}")
# Result: t = -4.56

# Calculate p-value (one-sided, left tail)
from scipy.stats import t
p_value = t.cdf(t_stat, df=n-1)
print(f"p-value: {p_value:.6f}")
# Result: p = 0.000005

H₀: p = 0.5 (50% have good sleep quality)
H₁: p < 0.5 (fewer than 50% have good sleep quality)

from scipy.stats import norm

# Count students with good sleep quality
good_sleep = (df['calidad_suenio'] == 'buena').sum()  # 27 students
n = len(df)  # 150
p_hat = good_sleep / n  # 0.18 (18%)

# Null hypothesis proportion
p_0 = 0.5

# Calculate z-statistic
z_stat = (p_hat - p_0) / np.sqrt(p_0 * (1 - p_0) / n)
print(f"z-statistic: {z_stat:.3f}")
# Result: z = -7.84

# Calculate p-value (left tail)
p_value = norm.cdf(z_stat)
print(f"p-value: {p_value:.10f}")
# Result: p ≈ 0.0000000000

H₀: μ_high = μ_low (no difference in stress)
H₁: μ_high < μ_low (high activity students have lower stress)

from scipy.stats import ttest_ind

# Separate groups
high_activity = df[df['nivel_actividad'] == 'alta']['estres_score']
low_activity = df[df['nivel_actividad'] == 'baja']['estres_score']

print(f"High activity: n={len(high_activity)}, mean={high_activity.mean():.2f}")
print(f"Low activity:  n={len(low_activity)}, mean={low_activity.mean():.2f}")
# High activity: n=26, mean=16.42
# Low activity:  n=45, mean=20.94

# Two-sample t-test (unequal variances)
t_stat, p_value_two_sided = ttest_ind(high_activity, low_activity, equal_var=False)

# Convert to one-sided test (H₁: μ_high < μ_low)
if t_stat < 0:
    p_value_one_sided = p_value_two_sided / 2
else:
    p_value_one_sided = 1 - p_value_two_sided / 2

print(f"\nt-statistic: {t_stat:.3f}")
print(f"p-value (two-sided): {p_value_two_sided:.4f}")
print(f"p-value (one-sided): {p_value_one_sided:.4f}")
# t = -3.15, p (one-sided) = 0.0014

H₀: μ_healthy = μ_poor (no difference in grades)
H₁: μ_healthy > μ_poor (healthy nutrition improves grades)

# Separate groups
healthy_nutrition = df[df['alimentacion_score'] >= 7]['promedio_notas']
poor_nutrition = df[df['alimentacion_score'] < 7]['promedio_notas']

# Check normality (visual)
import matplotlib.pyplot as plt
fig, axes = plt.subplots(1, 2, figsize=(12, 4))
axes[0].hist(healthy_nutrition, bins=15, edgecolor='black')
axes[0].set_title('Healthy Nutrition Group')
axes[1].hist(poor_nutrition, bins=15, edgecolor='black')
axes[1].set_title('Poor Nutrition Group')
plt.show()

# Both look approximately normal ✓

print("Healthy Nutrition Group (score ≥ 7):")
print(f"  n = {len(healthy_nutrition)}")
print(f"  Mean = {healthy_nutrition.mean():.2f}")
print(f"  SD = {healthy_nutrition.std(ddof=1):.2f}")

print("\nPoor Nutrition Group (score < 7):")
print(f"  n = {len(poor_nutrition)}")
print(f"  Mean = {poor_nutrition.mean():.2f}")
print(f"  SD = {poor_nutrition.std(ddof=1):.2f}")

from scipy.stats import ttest_ind

t_stat, p_value_two = ttest_ind(healthy_nutrition, poor_nutrition, equal_var=False)

# One-sided test (healthy > poor)
if t_stat > 0:
    p_value = p_value_two / 2
else:
    p_value = 1 - p_value_two / 2

print(f"\nTest Results:")
print(f"  t-statistic: {t_stat:.3f}")
print(f"  p-value (one-sided): {p_value:.4f}")

if p_value < 0.05:
    print(f"\nDecision: Reject H₀ (p = {p_value:.4f} < 0.05)")
    print("Conclusion: Students with healthy nutrition have significantly higher grades.")
else:
    print(f"\nDecision: Fail to reject H₀ (p = {p_value:.4f} ≥ 0.05)")
    print("Conclusion: No significant evidence that nutrition affects grades.")

# Cohen's d: standardized mean difference
mean_diff = healthy_nutrition.mean() - poor_nutrition.mean()
pooled_std = np.sqrt((healthy_nutrition.var(ddof=1) + poor_nutrition.var(ddof=1)) / 2)
cohens_d = mean_diff / pooled_std

print(f"\nEffect Size:")
print(f"  Raw difference: {mean_diff:.2f} points")
print(f"  Cohen's d: {cohens_d:.3f}")

# Interpret Cohen's d
if abs(cohens_d) < 0.2:
    interpretation = "negligible"
elif abs(cohens_d) < 0.5:
    interpretation = "small"
elif abs(cohens_d) < 0.8:
    interpretation = "medium"
else:
    interpretation = "large"
    
print(f"  Interpretation: {interpretation} effect")