The statement is **False**.
In a chi-square test for independence, the expected frequencies for each cell of a contingency table are not the same across all rows. The expected frequency for a cell within a contingency table is calculated based on the marginal totals (the sums of rows and columns) and the grand total (the sum of all observations).
The formula for calculating the expected frequency for any cell in the contingency table is:
\[ \text{Expected frequency} = \frac{(\text{Row total} \times \text{Column total})}{\text{Grand total}} \]
Thus, the expected frequency for a particular cell depends on the row and column in which it is located, meaning it will vary from row to row (and from column to column) unless all the marginal totals for rows and columns are equal. If the marginal totals are not equal across rows and columns, the expected frequencies in each row will differ.
For example, consider the following contingency table:
|\ | Column 1 | Column 2 | Row Total |
|------|----------|----------|-----------|
| Row 1| a | b | a + b |
| Row 2| c | d | c + d |
| Column Total | a + c | b + d | a + b + c + d (Grand Total) |
The expected frequency for the cell at Row 1, Column 1 would be:
\[ \text{Expected frequency (Row 1, Column 1)} = \frac{(a + b) \times (a + c)}{a + b + c + d} \]
While the expected frequency for the cell at Row 2, Column 1 would be:
\[ \text{Expected frequency (Row 2, Column 1)} = \frac{(c + d) \times (a + c)}{a + b + c + d} \]
These expected frequencies are not the same unless a + b equals c + d and a + c equals b + d.
Therefore, the expected frequencies for any row in a chi-square test for independence are not necessarily the same as the expected frequencies in every other row.
