Write a paragraph that describes the relationship between race (respondent's rac

Step 1
Here is the contingency table:
$\begin{array}{|cccc|}\hline \text{Race/Option}& \text{Favor}& \text{Oppose}& \text{Total}\\ \text{White}& 1063& 484& 1547\\ \text{African American}& 187& 91& 278\\ \text{Other}& 117& 64& 181\\ \text{Total}& 1367& 639& 2006\\ \hline\end{array}$
Let's calculate the row percentages to see whether there is any obvious retalionship between race and opinion on capital punishment.
Let ${\displaystyle n_{ij}}$ be the number in row i and column j in the contingency table, and let ${\displaystyle s_i}$ be the total row counts. Then the row percentages are calculated as let ${\displaystyle s_i}$ be the total row counts. Then the row percentages are calculated as
${\displaystyle p_{ij}=\frac{n_{ij}}{s_i}\cdot 100\mathrm{\%}}$
Here is the table of row percentages:
$\begin{array}{|cccc|}\hline \text{Race/Option}& \text{Favor}& \text{Oppose}& \text{Total}\\ \text{White}& \frac{1063}{1547}\cdot 100\mathrm{\%}=68.71\mathrm{\%}& \frac{484}{1547}\cdot 100\mathrm{\%}=31.29\mathrm{\%}& 100\mathrm{\%}\\ \text{African American}& \frac{187}{278}\cdot 100\mathrm{\%}=67.27\mathrm{\%}& \frac{91}{278}\cdot 100\mathrm{\%}=32.73\mathrm{\%}& 100\mathrm{\%}\\ \text{Other}& \frac{117}{181}\cdot 100\mathrm{\%}=64.64\mathrm{\%}& \frac{64}{181}\cdot 100\mathrm{\%}=35.36\mathrm{\%}& 100\mathrm{\%}\\ \text{Total}& \frac{1367}{2006}\cdot 100\mathrm{\%}=68.15\mathrm{\%}& \frac{639}{2006}\cdot 100\mathrm{\%}=31.85\mathrm{\%}& 100\mathrm{\%}\\ \hline\end{array}$
So, at the first glance it would seem as if there was no relationship between the two variables, since the rows have about the equal distribution of percentages, which indicates the independence between variables, i.e. no relationship.
Let's carry out the chi-square test to make sure. Notice that before we;ve only dealt with cases with ${\displaystyle 2\times 2}$ tables, but chi-square test can be applied to any (large) size of the table, and the procedure is exactly the same as with ${\displaystyle 2\times 2}$ tables!
Step 2
So, we will test:
${\displaystyle H_0:}$ Race and option on capital punishment are not related
${\displaystyle H_1:}$ Race and option on capital punishment are related
First we need the table of expected counts. In order to construct the table , consider the following:
The expected count for each cell is calculated as
${\displaystyle \text{Expected}\frac{\text{Total row count}\cdot \text{Total column count}}{\text{Sample size}}}$
Here is the table of expected counts:
$\begin{array}{|cccc|}\hline \text{Race/Option}& \text{Favor}& \text{Oppose}& \text{Total}\\ \text{White}& \frac{1547\cdot 1367}{2006}=1045.21& \frac{1547\cdot 639}{2006}=492.79& 1547\\ \text{African American}& \frac{278\cdot 1367}{2006}=189.44& \frac{278\cdot 639}{2006}=88.56& 278\\ \text{Other}& \frac{181\cdot 1367}{2006}=123.34& \frac{181\cdot 639}{2006}=57.66& 181\\ \text{Total}& 1367& 639& 2006\\ \hline\end{array}$
The value of chi-square statistics is
${\chi }^2=\sum \frac{{\text{Observed count-Expected counts}}^2}{\text{Expected count}}$
${\displaystyle =\frac{{(1063-1045.21)}^2}{1045.21}+\frac{{(484-492.79)}^2}{492.79}+\frac{{(187-189.44)}^2}{189.44}+}$
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