In a study of treatments for very painful "cluster" headaches, 147 patients were treated with oxygen and 152 other patients were given a placebo consisting of ordinary air. Among the 147 patients in the oxygen treatment group, 111 were free from headaches 15 minutes after treatment. Among the 152 patients given the placebo, 27 were free from headaches 15 minutes after treatment. Use a 0.05 significance level to test the claim that the oxygen treatment is effective. Complete parts (a) through (c) below. \[ \mathrm{z}= \] (Round to two decimal places as needed.)

Step 1 ：Given that 147 patients were treated with oxygen and 152 patients were given a placebo. Among the 147 patients in the oxygen treatment group, 111 were free from headaches 15 minutes after treatment. Among the 152 patients given the placebo, 27 were free from headaches 15 minutes after treatment.

Step 2 ：We are asked to test the claim that the oxygen treatment is effective using a 0.05 significance level. This is a problem of hypothesis testing for proportions. The null hypothesis is that the oxygen treatment is not effective, meaning the proportion of patients free from headaches after the treatment is the same for both groups. The alternative hypothesis is that the oxygen treatment is effective, meaning the proportion of patients free from headaches after the treatment is higher for the oxygen group than for the placebo group.

Step 3 ：We can use the z-test for proportions to test these hypotheses. The z-score is calculated as follows: \[ z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})} \] where \(\hat{p}_1\) and \(\hat{p}_2\) are the sample proportions of the oxygen and placebo groups respectively, \(\hat{p}\) is the pooled sample proportion, and \(n_1\) and \(n_2\) are the sample sizes of the oxygen and placebo groups respectively.

Step 4 ：The pooled sample proportion \(\hat{p}\) is calculated as follows: \[ \hat{p} = \frac{x_1 + x_2}{n_1 + n_2} \] where \(x_1\) and \(x_2\) are the number of successes (patients free from headaches) in the oxygen and placebo groups respectively.

Step 5 ：Substituting the given values into the formulas, we get: \[ x1 = 111, n1 = 147, x2 = 27, n2 = 152, \hat{p}_1 = 0.7551020408163265, \hat{p}_2 = 0.17763157894736842, \hat{p} = 0.46153846153846156 \]

Step 6 ：Calculating the z-score, we get: \[ z = 10.013671699957575 \]

Step 7 ：The z-score is approximately 10.01. This is a very high value, which indicates that the difference between the proportions of patients free from headaches in the oxygen and placebo groups is very significant. Therefore, we can reject the null hypothesis that the oxygen treatment is not effective. This means that the oxygen treatment is effective.

Step 8 ：Final Answer: The z-score is \(\boxed{10.01}\).