In a chi-square test for independence, how does the difference between the expected frequency $fe$ and the observed frequency $fo$ affect the value of the chi-square statistic?

• A. A larger difference between $fe$ and $fo$ increases the chi-square statistic, making it more likely to reject the null hypothesis.
• B. A larger difference between $fe$ and $fo$ decreases the chi-square statistic, making it less likely to reject the null hypothesis.
• C. The difference between $fe$ and $fo$ has no effect on the chi-square statistic.
• D. A smaller difference between $fe$ and $fo$ increases the chi-square statistic.

Answer: (A)

Verified step by step guidance

1. Recall the formula for the chi-square statistic in a test for independence: \[\chi^2 = \sum \frac{(f_o - f_e)^2}{f_e}\] where \(f_o\) is the observed frequency and \(f_e\) is the expected frequency for each cell in the contingency table.
2. Understand that the numerator \((f_o - f_e)^2\) represents the squared difference between observed and expected frequencies. This means that larger differences will increase the value of this term.
3. Since the chi-square statistic sums these squared differences divided by the expected frequencies, larger differences between \(f_o\) and \(f_e\) will contribute more to the overall chi-square value.
4. A higher chi-square statistic indicates a greater discrepancy between observed and expected data, which makes it more likely to reject the null hypothesis of independence.
5. Therefore, the difference between \(f_o\) and \(f_e\) directly affects the chi-square statistic: the larger the difference, the larger the chi-square value, increasing the chance of rejecting the null hypothesis.