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.