Implementing Common Hypothesis Tests in Python: Z‑test, t‑test, F‑test and Data Exploration

This article teaches how to perform common hypothesis tests with Python.

It explains that the choice of test depends on the underlying distribution: for example, when two sample variances follow an F‑distribution, the interval estimation uses F‑critical values and the F‑test is applied.

Python packages used

Statsmodels – statistical modeling and inference.

SciPy – scientific computing, including statistical functions.

Data import example (Iris dataset)

from sklearn.datasets import load_iris
import numpy as np
# Load Iris data
iris = load_iris()
iris = pd.DataFrame(iris.data, columns=['sepal_length','sepal_width','petal_legth','petal_width'])
print(iris)

One‑sample Z‑test on Iris petal length (null hypothesis: mean = 4.2)

import statsmodels.stats.weightstats
z, pval = statsmodels.stats.weightstats.ztest(iris['petal_legth'], value=4.2)
print(z, pval)  # Output: p < 0.05 → reject H0

One‑sample t‑test on Iris petal length (null hypothesis: mean = 4.0)

import scipy.stats
t, pval = scipy.stats.ttest_1samp(iris['petal_legth'], popmean=4.0)
print(t, pval)  # Output: p > 0.05 → fail to reject H0

Two‑sample t‑test (simulated)

# Split Iris data
iris_1 = iris[iris.petal_legth >= 2]
iris_2 = iris[iris.petal_legth < 2]
print(np.mean(iris_1['petal_legth']))
print(np.mean(iris_2['petal_legth']))
import scipy.stats
t, pval = scipy.stats.ttest_ind(iris_1['petal_legth'], iris_2['petal_legth'])
print(t, pval)  # p < 0.05 → reject H0

Exercise: Human temperature dataset

The dataset contains 130 rows with three columns: Gender (1 = male, 2 = female), Temperature, and HeartRate.

Load and describe data

import numpy as np
import pandas as pd
from scipy import stats
data = pd.read_csv('C:\\Users\\baihua\\Desktop\\test.csv')
print(data.head())
print(data.describe())
# Mean temperature ≈ 98.25°F

One‑sample Z‑test for mean temperature = 98.6°F

import statsmodels.stats.weightstats
z, pval = statsmodels.stats.weightstats.ztest(data['Temperature'], value=98.6)
print(z, pval)  # p < 0.05 → reject H0

Normality test (Kolmogorov‑Smirnov)

stats.kstest(data['Temperature'], 'norm')
# Result: statistic=1.0, pvalue=0.0 → not normal

Fit t‑distribution and perform KS test

np.random.seed(1)
ks = stats.t.fit(data['Temperature'])
df, loc, scale = ks
t_estm = stats.t.rvs(df=df, loc=loc, scale=scale, size=data.shape[0])
print(stats.ks_2samp(data['Temperature'], t_estm))  # p≈0.43 → cannot reject t‑fit

Fit chi‑square distribution and perform KS test

chi_sq = stats.chi2.fit(data['Temperature'])
df, loc, scale = chi_sq
chi_estm = stats.chi2.rvs(df=df, loc=loc, scale=scale, size=data.shape[0])
print(stats.ks_2samp(data['Temperature'], chi_estm))  # p≈0.40 → cannot reject chi‑square fit

Identify outliers using chi‑square quantiles (2.5% and 92.5%)

lower = chi_sq_distribution.ppf(0.025)
upper = chi_sq_distribution.ppf(0.925)
print(data['Temperature'][data['Temperature'] < lower])
print(data['Temperature'][data['Temperature'] > upper])

Gender temperature difference (two‑sample t‑test)

male = data[data['Gender'] == 1]['Temperature']
female = data[data['Gender'] == 2]['Temperature']
t, pval = scipy.stats.ttest_ind(male, female)
print(t, pval)  # p < 0.05 → significant difference

Correlation between temperature and heart rate

stat, p = stats.pearsonr(data['HeartRate'], data['Temperature'])
print('stat=%.3f, p=%.3f' % (stat, p))  # stat≈0.004 → no correlation

Figures illustrating distributions and scatter plots are included in the original article.