Problems

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Chi-Square Test

Probability and Statistics

Medium

Given an $r\times c$ contingency table of observed counts $O$ , compute each expected count under independence:

E_{ij} = \frac{R_iC_j}{N}

Then compute the chi-square statistic:

\chi^2 = \sum_{i=1}^{r}\sum_{j=1}^{c}\frac{(O_{ij}-E_{ij})^2}{E_{ij}}

Here, $R_i$ is row $i$ 's total, $C_j$ is column $j$ 's total, and $N$ is the grand total. Return chi2 as a Python float and expected as a NumPy array in a dictionary.

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Examples

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

Output: {"chi2": 6.666667, "expected": [[15.0, 15.0], [15.0, 15.0]]}

Explanation: Equal row and column totals produce expected counts of 15 in every cell.

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

Output: {"chi2": 0.0, "expected": [[20.0, 30.0], [40.0, 60.0]]}

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

Output: {"chi2": 0.0, "expected": [[25.0, 25.0], [25.0, 25.0]]}

Hint 1

Use np.outer(row_totals, column_totals) / total for expected counts.

Hint 2

Sum (C - expected) ** 2 / expected across the complete table.

Requirements

Compute row totals, column totals, and the grand total
Build the expected table with an outer product
Return exactly chi2 and expected in a dictionary
expected must be a NumPy array

Constraints

C is a nonempty two-dimensional table of positive counts
Every expected frequency is positive
Use NumPy only

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