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03-two-samples-two-means-z.py
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03-two-samples-two-means-z.py
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import matplotlib.pyplot as plt
import matplotlib.ticker as ticker
from scipy import stats
import numpy as np
import pandas as pd
import math
ALPHA = 0.05
NUMBER_OF_TESTS = 1000
POP_1_MEAN = 10 # 10 15
POP_1_STD = 2 # 2 4
SAMPLE_1_SIZE = 10
POP_2_MEAN = 10
POP_2_STD = 4
SAMPLE_2_SIZE = 5
POP_1_PROB_DENS_X = np.linspace(POP_1_MEAN - 4*POP_1_STD, POP_1_MEAN + 4*POP_1_STD, 1000)
# https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html
POP_1_PROB_DENS_Y = stats.norm.pdf(POP_1_PROB_DENS_X, loc = POP_1_MEAN, scale = POP_1_STD)
POP_1_PROB_MAX = stats.norm.pdf(POP_1_MEAN, loc = POP_1_MEAN, scale = POP_1_STD)
POP_2_PROB_DENS_X = np.linspace(POP_2_MEAN - 4*POP_2_STD, POP_2_MEAN + 4*POP_2_STD, 1000)
# https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html
POP_2_PROB_DENS_Y = stats.norm.pdf(POP_2_PROB_DENS_X, loc = POP_2_MEAN, scale = POP_2_STD)
POP_2_PROB_MAX = stats.norm.pdf(POP_2_MEAN, loc = POP_2_MEAN, scale = POP_2_STD)
fig, ax = plt.subplots(1, 1)
ax.grid(axis='both', linestyle='--', color='0.95')
ax.xaxis.set_major_locator(ticker.MultipleLocator(1.0))
ax.set_xlim(POP_2_MEAN - 4.2*POP_2_STD, POP_2_MEAN + 4.2*POP_2_STD)
ax.yaxis.set_major_locator(ticker.MultipleLocator(0.01))
text0 = ax.text(POP_2_MEAN + 2*POP_2_STD, POP_2_PROB_MAX * 0.3, f'')
text_x_dash_1 = ax.text(0, 0, f'x̄1')
text_x_dash_2 = ax.text(0, 0, f'x̄2')
text_x_dash_diff = ax.text(0, 0, f'x̄1-x̄2')
text_mu_1 = ax.text(POP_1_MEAN, -0.008, f'μ1=μ2')
# text_mu_2 = ax.text(POP_2_MEAN, -0.008, f'μ2')
text_mu_diff = ax.text(POP_1_MEAN - POP_2_MEAN, -0.004, f'μ1-μ2')
dots0, = ax.plot([], [], 'bo', alpha=1.0)
dots1, = ax.plot([], [], 'go', alpha=1.0)
vlines0 = ax.vlines([], [], [], color='r', alpha=1.0)
vlines1 = ax.vlines([], [], [], color='r', alpha=1.0)
vlines2 = ax.vlines([], [], [], color='r', alpha=1.0)
vlines3 = ax.vlines([], [], [], color='r', alpha=1.0)
fill = ax.fill([], [], alpha=0.4, hatch="X", color='lightblue')
# Distributions
ax.plot(POP_1_PROB_DENS_X, POP_1_PROB_DENS_Y, color='blue')
ax.plot(POP_2_PROB_DENS_X, POP_2_PROB_DENS_Y, color='green')
# Population Mean
ax.vlines([POP_1_MEAN], [0], [POP_1_PROB_MAX], color='red', linestyle='dashed', linewidth=3)
ax.vlines([POP_2_MEAN], [0], [POP_2_PROB_MAX], color='red', linestyle='dashed', linewidth=3)
ax.vlines([POP_1_MEAN - POP_2_MEAN], [0], [min(POP_1_PROB_MAX, POP_2_PROB_MAX)], color='red', linestyle='dashed', linewidth=3)
h0_counter = 0
h1_counter = 0
for x in range(NUMBER_OF_TESTS):
# https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html
sample1 = stats.norm.rvs(size=SAMPLE_1_SIZE, loc = POP_1_MEAN, scale = POP_1_STD)
s1_mean = np.mean(sample1)
s1_std = np.std(sample1)
# https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html
sample2 = stats.norm.rvs(size=SAMPLE_2_SIZE, loc = POP_2_MEAN, scale = POP_2_STD)
s2_mean = np.mean(sample2)
s2_std = np.std(sample2)
# https://en.wikipedia.org/wiki/97.5th_percentile_point
# Significance Level (α): 0.05
# Z-Value (two-tailed): +/- 1.96
# z_005 = 1.96
# https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html
z_alpha = stats.norm.ppf(1 - ALPHA/2) # alpha divide by 2 - two-tailed test
# Using Populations Std Deviations
moe = z_alpha * math.sqrt(POP_1_STD**2/SAMPLE_1_SIZE + POP_2_STD**2/SAMPLE_2_SIZE)
if ((s1_mean - s2_mean) - moe) <= (POP_1_MEAN - POP_2_MEAN) \
and (POP_1_MEAN - POP_2_MEAN) <= ((s1_mean - s2_mean) + moe):
h0_counter += 1
else:
h1_counter += 1
if x < 50 or x == NUMBER_OF_TESTS - 1:
text0.set_text(
f'Significance Level (α): {ALPHA * 100:.2f} % \n'
+ f'Z({1 - ALPHA/2:.3f}) Two-Tailed: {z_alpha:.6f} \n\n'
+ f'Population Mean (μ1): {POP_1_MEAN:.4f} \n'
+ f'Population Mean (μ2): {POP_2_MEAN:.4f} \n'
+ f'Population Standard Deviation (σ1): {POP_1_STD:.4f}\n'
+ f'Population Standard Deviation (σ2): {POP_2_STD:.4f}\n\n'
+ f'Sample Size 1 (n1): {SAMPLE_1_SIZE}\n'
+ f'Sample Size 2 (n2): {SAMPLE_2_SIZE}\n'
+ f'Sample Mean 1 (x̄1): {s1_mean:.4f} \n'
+ f'Sample Mean 2 (x̄2): {s2_mean:.4f} \n'
+ f'Sample Standard Deviation 1 (s1): {s1_std:.4f}\n'
+ f'Sample Standard Deviation 2 (s2): {s2_std:.4f}\n\n'
+ f'H0 (μ1-μ2={POP_1_MEAN - POP_2_MEAN:.2f}) is TRUE: {h0_counter} (Correct)\n'
+ f'H1 (μ1-μ2≠{POP_1_MEAN - POP_2_MEAN:.2f}) is TRUE: {h1_counter} (False positive)\n\n'
+ f'Actuall Type I Error Percent: {100 * h1_counter / (h0_counter + h1_counter):.2f} %'
)
text_x_dash_1.set_position((s1_mean, POP_1_PROB_MAX))
text_x_dash_2.set_position((s2_mean, POP_2_PROB_MAX))
text_x_dash_diff.set_position((s1_mean-s2_mean, min(POP_1_PROB_MAX, POP_2_PROB_MAX)))
# Sample Means
vlines0.remove()
vlines0 = ax.vlines([s1_mean], [0], [POP_1_PROB_MAX], color='blue', linestyle='dashed', linewidth=3)
vlines1.remove()
vlines1 = ax.vlines([s2_mean], [0], [POP_2_PROB_MAX], color='green', linestyle='dashed', linewidth=3)
vlines2.remove()
vlines2 = ax.vlines([s1_mean - s2_mean], [0], [min(POP_1_PROB_MAX, POP_2_PROB_MAX)], color='black', linestyle='dashed', linewidth=3)
# Dots
dots0.set_data(sample1, stats.norm.pdf(sample1, loc = POP_1_MEAN, scale = POP_1_STD))
dots1.set_data(sample2, stats.norm.pdf(sample2, loc = POP_2_MEAN, scale = POP_2_STD))
# Confidence Interval
vlines3.remove()
vlines3 = ax.vlines([
(s1_mean - s2_mean) - moe,
(s1_mean - s2_mean) + moe
], [
0,
0
], [
min(POP_1_PROB_MAX, POP_2_PROB_MAX),
min(POP_1_PROB_MAX, POP_2_PROB_MAX)
], color='blue', linestyle='dashed', linewidth=1)
fill[0].remove()
fill = ax.fill([
(s1_mean - s2_mean) - moe,
(s1_mean - s2_mean) - moe,
(s1_mean - s2_mean) + moe,
(s1_mean - s2_mean) + moe
], [
0,
min(POP_1_PROB_MAX, POP_2_PROB_MAX),
min(POP_1_PROB_MAX, POP_2_PROB_MAX),
0
], alpha=0.4, hatch="//", color='lightblue')
plt.tight_layout()
plt.pause(0.5)
plt.tight_layout()
plt.show()