Problems

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Chi-Square Test

Probability and Statistics

Medium

Given an r×c contingency table C, compute the chi-square test statistic and expected frequency table for testing independence.

Chi-Square Test for Independence:

Test Statistic:

\chi^2 = \sum \frac{(O - E)^2}{E}

Expected Frequencies:

E_{ij} = \frac{\text{row}_i \cdot \text{col}_j}{\text{total}}

Function Arguments

C: 2D array - Contingency table (observed frequencies)

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Examples

Input: C=[[10,20],[20,10]]

Output: chi2=6.667, expected=[[15,15],[15,15]]

Input: C=[[20,30],[40,60]]

Output: chi2=0.0, expected=[[20,30],[40,60]]

Input: C=[[25,25],[25,25]]

Output: chi2=0.0, expected=[[25,25],[25,25]]

Hint 1

Use np.sum() with axis parameter for row/column totals.

Hint 2

Use np.outer() to compute expected frequencies matrix.

Hint 3

Chi-square: np.sum((C - expected) ** 2 / expected).

Requirements

Return tuple: (chi2, expected)
chi2: scalar float
expected: NumPy array (same shape as C)
Vectorized computation (no nested loops)

Constraints

C is 2D array with positive integers
NumPy only; time limit: 300ms

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