Step 1 :The null hypothesis in this case is that the proportion of treated subjects who experienced headaches is less than 10%.
Step 2 :The P-value is the probability that we would observe a result as extreme as, or more extreme than, the result we actually observed, given that the null hypothesis is true.
Step 3 :In this case, the P-value is 0.5789487151, which is greater than the significance level of 0.01.
Step 4 :This means that we would expect to see a result as extreme as, or more extreme than, the one we observed more than 5.78% of the time if the null hypothesis were true.
Step 5 :Because the P-value is greater than the significance level, we do not have sufficient evidence to reject the null hypothesis.
Step 6 :Therefore, the correct answer is C. Fail to reject the null hypothesis because the P-value is greater than the significance level, α.
Step 7 :Final Answer: \(\boxed{\text{C. Fail to reject the null hypothesis because the P-value is greater than the significance level, } \alpha}\).