Question

# Find the expected count and the contribution to the chi-square statistic for the (Group 1, No)

Two-way tables

Find the expected count and the contribution to the chi-square statistic for the (Group 1, No) cell in the two-way table below.

$$\begin{array}{|c|c|c|}\hline&\text{Yes}&\text{No}&\text{Total}\\\hline\text{Group 1} &56 & 42 & 98\\ \hline \ \text{Group 2}&135&67&202 \\ \hline \text{Group 3}&66&23&89 \\ \hline \text{Total}&257&132&389 \\ \hline \end{array}$$
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 = ?

2021-01-17
Step 1
Observed values with marginal totals:
$$\begin{array}{|c|c|c|}\hline&\text{Yes}&\text{No}&\text{Total}\\\hline\text{Group 1} &56 & 42 & 98\\ \hline \ \text{Group 2}&135&67&202 \\ \hline \text{Group 3}&66&23&89 \\ \hline \text{Total}&257&132&389 \\ \hline \end{array}$$
Step 2
Expected values:
Expected values are the product of the column and row total divided by the table total. $$\begin{array}{|c|c|c|}\hline&\text{Yes}&\text{No}\\\hline\text{Group 1} &\frac{257\times98}{389}=64.7 & \frac{132\times98}{389}=33.2\\ \hline \text{Group 2}&\frac{257\times202}{389}=133.5&\frac{132\times202}{389}=68.5 \\ \hline \text{Group 3}&\frac{257\times89}{389}=58.8&\frac{132\times89}{389}=30.2 \\ \hline \end{array}$$
Step 3
The expected count for (Group 1, No) is as follows:
$$\text{Expected count}=\frac{132 \times 98}{389}$$
$$=33.2545$$
$$\approx33.3$$ The contribution to test statistic is as follows: $$\text{contribution to chi-square}=\frac{(O-E)^2}{E}$$
$$=\frac{(42-33.2545)^2}{33.2545}$$
$$=2.29995$$
$$\approx2.300$$