Discussion
Consider the \(2 \times 2\) contingency table on two attributes A and B :
| \(A_{1}\) | \(A_{2}\) | |
| \(B_{1}\) | 10 | 20 |
| \(B_{2}\) | 30 | 40 |
What is the value of \(\chi^{2}\) for testing the independence of the attributes A and B ?
Answer: B
The value of \(\chi^{2}\) is 0.794.
For a \(2 \times 2\) table with cells a = 10, b = 20, c = 30, d = 40 and N = 100, the shortcut formula is
\[ \chi^{2} = \frac{N(ad - bc)^{2}}{(a+b)(c+d)(a+c)(b+d)} \]
Here \(ad - bc = 400 - 600 = -200\), the row totals are 30 and 70, and the column totals are 40 and 60:
\[ \begin{aligned} \chi^{2} &= \frac{100 \times 40000}{30 \times 70 \times 40 \times 60} \\ &= \frac{4000000}{5040000} \approx 0.794 \end{aligned} \]
The same value comes from \(\sum (O - E)^{2}/E\) with expected counts 12, 18, 28 and 42. The other values do not agree with either formula. With 1 degree of freedom, 0.794 is far below the 5% point 3.84, so independence is not rejected.
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