2022-02-20

Find the expected count and the contribution to the chi-square statistic for the (Treatment, Disagree) cell in the two-way table below.

Strongly Agree Agree Neutral Disagree Strongly Disagree 35 48 3 14 9 56 43 11 5 0

Vasquez

Solution

The row and column totals are,

$\begin{array}{|ccccccc|}\hline & \text{Strongly Agree}& \text{Agree}& \text{Neutral}& \text{Disagree}& \text{Strongly Disagree}& \text{Total}\\ \text{Control}& 35& 48& 3& 14& 9& 109\\ \text{Treatment}& 56& 43& 11& 5& 0& 115\\ \text{Total}& 91& 91& 14& 19& 9& 224\\ \hline\end{array}$

The expected count for the (Treatment, Disagree) cell is,

$\text{Expected count}=\frac{\left(\text{Row total}\right)\left(\text{Column totalt}\right)}{\text{Grand total}}\phantom{\rule{0ex}{0ex}}=\frac{115×19}{224}\phantom{\rule{0ex}{0ex}}\frac{2185}{224}\phantom{\rule{0ex}{0ex}}=9.7545\phantom{\rule{0ex}{0ex}}\approx 9.8$

Thus, the expected count for the (Treatment, Disagree) cell is $9.8$. The contribution to the chi-square statistic for the (Treatment, Disagree) cell is,

$\text{Contribution}=\frac{\left(\text{Observed-Expected}{\right)}^{2}}{\text{Expected}}\phantom{\rule{0ex}{0ex}}=\frac{\left(5-9.7545\right)}{9.7545}\phantom{\rule{0ex}{0ex}}\frac{22.60525025}{9.7545}\phantom{\rule{0ex}{0ex}}=2.317$

Thus, the contribution to the chi-square statistic for the (Treatment, Disagree) cell is $2.317$.

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