Test auf Normalverteilung
Test auf Normalverteilung¶
Generieren Sie in 100.000 (gleichwahrscheinliche) Würfe eines Würfels und berechnen Sie Mittelwert und Standardabweichung der Würfelsumme. Wählen Sie aus den Würfelwürfen 200 Stichproben mit unterschiedlichen Stichprobenumfängen. Ab welchem Stichprobenumfang (3, 5, 7, 10, 15, 20, 30, 50) können wir dafon ausgehen, dass die Stichprobenverteilung des Mittelwertes normalverteilt ist. Nutzen sie zur Validierung des Hypothese den Wilk-Shapiro Test.
Hilfsfunktionen
import numpy as np
from scipy import stats
def test_for_normal_distribution(x, verbose=True):
"""Function to test if a sample is normally distributed.
Therefore the Shapiro-Wilk test is employed. If the p-value is <0.05 we recject the null hypothesis and hence
conclude that the data is not normally distrubuted for reference see
https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.shapiro.html"""
shapiro_test = stats.shapiro(x)
pvalue = shapiro_test.pvalue
if verbose:
print(f"p-value: {pvalue}")
if pvalue < 0.05:
print(
f"The null hypothesis is rejected, the data is NOT normally distributed."
)
else:
print(
f"Given the data the null hypothesis cannot be rejected, the data is likely normally distributed."
)
return pvalue
def dice_roll(nrolls: int, nsides: int = 6, seed=None) -> list:
"""Function to simulate a dice roll
params:
nrolls: number of rolls/dices
nsides: number of sides
"""
if seed is not None:
np.random.seed(seed)
return [np.random.randint(1, nsides + 1) for x in range(nrolls)]
# Frage 1 ...
# experiment
N = 100000
seed = 42
experiment = dice_roll(N, seed=seed)
# Validierung
for n in [3, 5, 7, 10, 15, 20, 30, 50]:
sample_means = []
for i in range(200):
sample = np.random.choice(experiment, n, replace=True)
sample_means.append(np.mean(sample))
print(f"\nSample size: {n}")
pvalue = test_for_normal_distribution(sample_means)
Sample size: 3
p-value: 0.011156341060996056
The null hypothesis is rejected, the data is NOT normally distributed.
Sample size: 5
p-value: 0.017443347722291946
The null hypothesis is rejected, the data is NOT normally distributed.
Sample size: 7
p-value: 0.1828618347644806
Given the data the null hypothesis cannot be rejected, the data is likely normally distributed.
Sample size: 10
p-value: 0.11042634397745132
Given the data the null hypothesis cannot be rejected, the data is likely normally distributed.
Sample size: 15
p-value: 0.10826049000024796
Given the data the null hypothesis cannot be rejected, the data is likely normally distributed.
Sample size: 20
p-value: 0.40129151940345764
Given the data the null hypothesis cannot be rejected, the data is likely normally distributed.
Sample size: 30
p-value: 0.5366109013557434
Given the data the null hypothesis cannot be rejected, the data is likely normally distributed.
Sample size: 50
p-value: 0.6726436614990234
Given the data the null hypothesis cannot be rejected, the data is likely normally distributed.