Question

# Find the expected count and the contribution to the chi-square statistic for the (Control, Disagree) cell in the two-way table below. begin{array}{|c|

Two-way tables
Find the expected count and the contribution to the chi-square statistic for the (Control, Disagree) cell in the two-way table below. $$\begin{array}{|c|c|c|}\hline&\text{Strongly Agree}&\text{Agree}&\text{Neutral}&\text{Disagree}&\text{Strongly Disagree}\\\hline\text{Control} &38&47&2&12&11\\ \hline \text{Treatment}&60&45&9&4&2 \\ \hline \end{array}\\$$
Round your answer for the excepted count to one decimal place, and your answer for the contribution to the chi-square statistic to three decimal places.
Expected count ?
Contribution to the chi-square statistic ?

2021-01-03
Step 1
Introduction:
The row and column totals are obtained as follows, for the given observed frequencies: $$\begin{array}{|c|c|c|}\hline&\text{Strongly Agree}&\text{Agree}&\text{Neutral}&\text{Disagree}&\text{Strongly Disagree}&\text{Total}\\\hline\text{Control} &38&47&2&12&11&110\\ \hline \text{Treatment}&60&45&9&4&2&120 \\ \hline \text{Total}&98&92&11&16&13&230 \\\hline \end{array}\\$$
Step 2
Calculation:
Consider the observation in the ith row and jth column of the table, that is, in the cell (i, j). Then, the expected frequency of the cell (i, j), if it is assumed that the two categories are independent, is $$\frac{[\text{(total of row i)} \cdot \text{(total of column j)}]}{\text{(grand total)}}$$
$$=\frac{(110 \cdot 16)}{230}$$
$$\approx 7.7$$
Thus, the expected count for the cell (Control, Disagree) is 7.7.
Now, the formula for the chi square test statistic is:
$$x^2=\sum_i\sum_j \frac{(O_{ij}-E_{ij})^2}{E_{ij}}$$
where
$$O_{ij} \text{is the observed frequency in cell} (i,j)$$
$$E_{ij} \text{is the expected frequency in cell} (i,j)$$
$$\frac{(O_{ij}-E_{ij})^2}{E_{ij}}$$
For the cell (Control, Disagree), that is, for the observation in row 1 and column 4, the component of the test statistic will be:
$$\frac{(O_{ij}-E_{ij})^2}{E_{ij}}$$
$$=\frac{(12-7.7)^2}{7.7}$$
$$\approx 2.416$$
Thus, the contribution to the chi-square statistic for the cell (Control, Disagree) is 2.416.