Submit quiz In a clinical trial of 2198 subjects treated with a certain drug, 22 reported headaches. In a control group of 1571 subjects given a placebo, 18 reported headaches. Denoting the proportion of headaches in the treatment group by $p_{t}$ and denoting the proportion of headaches in the control (placebo) group by $p_{0}$, the relative risk is $p_{t} / p_{c}$. The relative risk is a measure of the strength of the effect of the drug treatment. Another such measure is the odds ratio, which is the ratio of the odds in favor of a headache for the treatment group to the odds in favor of a headache for the control (placebo) group, found by evaluating $\frac{p_{t}^{\prime}\left(1-p_{t}\right)}{p_{c}^{\prime}\left(1-p_{c}\right)}$. The relative risk and odds ratios are commonly used in medicine and epidemiological studies. Find the relative risk and odds ratio for the headache data. What do the results suggest about the risk of a headache from the drug treatment? Find the odds ratio for the headache data. The odds ratio $=\square$ (Round to three decimal places as needed. $)$

Step 1 ：Given that there are 2198 subjects treated with a certain drug and 22 of them reported headaches, we can calculate the proportion of headaches in the treatment group, denoted as \(p_{t}\), by dividing the number of subjects who reported headaches by the total number of subjects in the treatment group. This gives us \(p_{t} = \frac{22}{2198} = 0.010009099181073703\).

Step 2 ：Similarly, given that there are 1571 subjects in the control group and 18 of them reported headaches, we can calculate the proportion of headaches in the control group, denoted as \(p_{c}\), by dividing the number of subjects who reported headaches by the total number of subjects in the control group. This gives us \(p_{c} = \frac{18}{1571} = 0.011457670273711012\).

Step 3 ：We can then calculate the odds ratio, which is a measure of the strength of the effect of the drug treatment, using the formula \(\frac{p_{t}(1-p_{t})}{p_{c}(1-p_{c})}\). Substituting the values of \(p_{t}\) and \(p_{c}\) we calculated earlier, we get the odds ratio = \(\frac{0.010009099181073703(1-0.010009099181073703)}{0.011457670273711012(1-0.011457670273711012)} = 0.8748520320732456\).

Step 4 ：Rounding to three decimal places, the odds ratio for the headache data is \(\boxed{0.875}\).