Two-Sample Tests and Chi-Square Analysis

Two-Sample Tests and Chi-Square Analysis

When you need to compare two groups or test whether two categorical variables are related, two workhorses of inferential statistics cover most situations: the independent two-sample t-test and the chi-square test of independence. Both appear regularly in business dashboards, policy reports, and data science projects — and both are tested on CompTIA Data+, the Google Data Analytics certificate, and IBM Data Science Professional assessments.

The Independent Two-Sample t-Test

The independent two-sample t-test asks: do two separate groups have the same population mean? For example, does City Department A resolve service requests in the same average time as City Department B?

Hypotheses

• Null hypothesis (H₀): The population means of the two groups are equal (μ₁ = μ₂).
• Alternative hypothesis (H₁): The population means differ (μ₁ ≠ μ₂) for a two-tailed test, or one is larger for a one-tailed test.

You choose a significance level — commonly α = 0.05 — before running the test. If the p-value returned by the test is less than α, you reject H₀.

Assumptions You Must Check

1. Independence: Observations in each group must not influence the other group. Department A's response times should not overlap with Department B's records.

2. Approximate normality: Each group's data should be roughly normally distributed, or the sample sizes should be large enough (often cited as n ≥ 30 per group) for the Central Limit Theorem to apply. A histogram or Q-Q plot helps you check this visually.

3. Equal (or unequal) variances: `scipy.stats.levene()` performs Levene's test. If Levene's p-value is below α, the variances differ significantly and you should pass `equal_var=False` to `scipy.stats.ttest_ind()`, which uses Welch's t-test instead of Student's t-test. Welch's version is generally safer and is the recommended default.

Running the Test in Python

```python

from scipy import stats

t_stat, p_value = stats.ttest_ind(dept_a, dept_b, equal_var=False)

print(f"t = {t_stat:.3f}, p = {p_value:.4f}")

```

If `p_value < 0.05`, you reject H₀ and conclude the mean response times differ. Always report the direction of the difference (which group was faster) and the practical size of that difference, not just the p-value.

The Chi-Square Test of Independence

The chi-square test of independence asks: are two categorical variables associated, or are they independent? For example, does the type of customer complaint (billing, delivery, product quality) depend on the customer's region?

The Contingency Table

You arrange counts in a contingency table with rows representing one variable and columns representing the other. Each cell holds the observed count for that combination.

Expected Frequencies

Under H₀ (the variables are independent), the expected count for a cell is:

> Expected = (Row Total × Column Total) / Grand Total

This formula embodies the assumption that if the variables are truly independent, each cell should hold exactly the proportion you would predict from the marginals alone. You must verify that no expected cell count falls below 5 — a key assumption of the chi-square approximation. If cells are too sparse, combine categories or use Fisher's exact test.

Computing the Chi-Square Statistic

The chi-square statistic sums the standardized squared deviations between observed (O) and expected (E) counts across all cells:

> χ² = Σ [(O − E)² / E]

A larger χ² means the observed data deviate more from what independence predicts.

Running the Test in Python

```python

from scipy.stats import chi2_contingency

import numpy as np

table = np.array([[34, 20, 15],

[18, 42, 10],

[12, 25, 30]])

chi2, p, dof, expected = chi2_contingency(table)

print(f"χ² = {chi2:.3f}, p = {p:.4f}, df = {dof}")

print("Expected:\n", expected)

```

If `p < 0.05`, you reject H₀ and conclude the two variables are associated.

Writing the Interpretation

A correct written interpretation has four parts:

1. State the test used and the α level chosen.

2. Report the test statistic and p-value.

3. State the decision (reject or fail to reject H₀).

4. Give a plain-language conclusion that answers the original business question.

Example: "Using an independent two-sample t-test at α = 0.05, we found t = −3.12, p = 0.002. We reject H₀ and conclude that Department A resolves requests significantly faster than Department B on average."

Data Accuracy and Integrity

Before running any test, verify your dataset for duplicate rows, missing values, and data-entry errors. A test run on dirty data produces misleading results that can misinform policy decisions — a real professional and ethical concern. Document every cleaning step in your notebook so the analysis is reproducible.

Lab Safety Note

This is a software lab. Follow your program's acceptable-use policy for shared computing resources. Do not upload identifiable personal data to public notebooks or cloud services without verifying that privacy requirements are met. Handle open datasets responsibly and cite your sources.