Question

${\chi }^2$ test statistic =

Short answer

The value of $X^2$test statistics is $2016.13$.

Step-by-step solution

Given Information

The following data are real. The cumulative number of AIDS cases reported for Santa Clara County is broken down by ethnicity as in Table 11.29.

$\begin{array}{|ll|}\hline \text{Ethnicity}& \text{Number of Cases}\\ \text{White}& 2,229\\ \text{Hispanic}& 1,157\\ \text{Black/African-American}& 457\\ \text{Asian, Pacific Islander}& 232\\ & \text{Total}=4,075\\ \hline\end{array}$

The percentage of each ethnic group in Santa Clara County is as in Table 11.30.

$\begin{array}{|lll|}\hline \text{Ethnicity}& \begin{array}{l}\text{Percentage of total county}\\ \text{population}\end{array}& \begin{array}{l}\text{Number expected (round to two}\\ \text{decimal places)}\end{array}\\ \text{White}& 42.9\%& 1748.18\\ \text{Hispanic}& 26.7\%& \\ \begin{array}{l}\text{BlacklAfrican-}\\ \text{American}\end{array}& 2.6\%& \\ \begin{array}{l}\text{Asian, Pacific}\\ \text{Islander}\end{array}& 27.8\%& \\ & \text{Total}=100\%& \\ \hline\end{array}$

Explanation

The number of cases per ethnic group is calculated as:$\begin{array}{|lll|}\hline \text{Ethnicity}& \begin{array}{l}\text{Percentage of total county}\\ \text{population}\end{array}& \begin{array}{l}\text{Number Expected(round to two decimal}\\ \text{places)}\end{array}\\ \text{White}& 42.9\%& 1748.18\\ \text{Hispanic}& 26.7\%& \frac{26.7}{100}\times 4075=1088.03\\ \begin{array}{l}\text{Black / African-}\\ \text{American}\end{array}& 2.6\%& \frac{2.6}{100}\times 4075=105.95\\ \text{Asian, Pacific Islander}& 27.8\%& \frac{27.8}{100}\times 4075=1132.85\\ \hline\end{array}$$\begin{array}{|lll|}\hline \text{Ethnicity}& \begin{array}{l}\text{Percentage of total county}\\ \text{population}\end{array}& \begin{array}{l}\text{Number Expected(round to two decimal}\\ \text{places)}\end{array}\\ \text{White}& 42.9\%& 1748.18\\ \text{Hispanic}& 26.7\%& \frac{26.7}{100}\times 4075=1088.03\\ \begin{array}{l}\text{Black / African-}\\ \text{American}\end{array}& 2.6\%& \frac{2.6}{100}\times 4075=105.95\\ \text{Asian, Pacific Islander}& 27.8\%& \frac{27.8}{100}\times 4075=1132.85\\ \hline\end{array}$

Explanation

We'll now use the steps below to calculate the $X^2$test statistic.

In the first column of the excel sheet, we write the expected frequencies.

2. In the first column of the excel sheet, record the observed frequencies.

3. Add a column to your spreadsheet with the heading (Observed - Expected)

4. Create a new column to calculate the square of the original (Observed - Expected)

5. Divide the square of (Observed - Predicted) by the expected frequency of each observation in the last column.

6. Finally, add the last two columns together.

Now, in Excel, perform the following calculation: