A new drug used as an aid for those who ant to quit smoking is tested. The adverse reaction of nausea has been studied in clinical trials, and the table summarizes results. Find the test statistic needed to test the claim that nausea is independent of whether the subject took a placebo or the new drug. Nausea 10,145
No nausea 795, 122
To test the claim that nausea is independent of whether the subject took a placebo or the new drug, we can use the chi-squared test for independence.
First, we need to set up a contingency table:
\[
\begin{array}{|c|c|c|}
\hline
& \text{Nausea} & \text{No Nausea} & \text{Total} \\
\hline
\text{Placebo} & 10 & 145 & 155 \\
\hline
\text{New Drug} & 795 & 122 & 917 \\
\hline
\text{Total} & 805 & 267 & 1072 \\
\hline
\end{array}
\]
Next, we calculate the expected values for each cell in the contingency table under the assumption of independence. To calculate the expected value for each cell, we use the formula:
\[ E = \frac{(\text{row total})(\text{column total})}{\text{grand total}} \]
For example, the expected value for the "Placebo - Nausea" cell would be:
\[ E_{\text{Placebo - Nausea}} = \frac{155 \times 805}{1072} = 116.34 \]
After calculating the expected values for all cells, we can then calculate the chi-squared test statistic using the formula:
\[ \chi^2 = \sum \frac{(O - E)^2}{E} \]
where \( O \) is the observed value and \( E \) is the expected value for each cell.
Finally, we compare the calculated chi-squared value to the critical value from the chi-squared distribution with \( (r-1) \times (c-1) \) degrees of freedom (where \( r \) is the number of rows and \( c \) is the number of columns) at a certain level of significance (e.g., 0.05) to determine if the variables are independent or not.