Question

Some commercial airplanes recirculate approximately \(50 \%\) of the cabin air in order to increase fuel efficiency. The authors of the paper "Aircraft Cabin Air Recirculation and Symptoms of the Common Cold" (Journal of the American Medical Association [2002]: \(483-486\) ) studied 1100 airline passengers who flew from San Francisco to Denver between January and April 1999. Some passengers traveled on airplanes that recirculated air and others traveled on planes that did not recirculate air. Of the 517 passengers who flew on planes that did not recirculate air, 108 reported post-flight respiratory symptoms, while 111 of the 583 passengers on planes that did recirculate air reported such symptoms. Is there sufficient evidence to conclude that the proportion of passengers with post-flight respiratory symptoms differs for planes that do and do not recirculate air? Test the appropriate hypotheses using \(\alpha=.05\). You may assume that it is reasonable to regard these two samples as being independently selected and as representative of the two populations of interest.

Short answer

Without calculations of the test-statistic and comparison to the critical value, no final short answer can be given. However, the method of comparing the calculated test statistic (z score) with the critical value will give a concrete answer whether there is sufficient evidence to state a significant difference in the proportion of passengers with respiratory symptoms between planes that do and do not recirculate air or not.

Step-by-step solution

State the Hypotheses

The null hypothesis (\(H_0\)) claims there is no difference in the proportion of passengers with post-flight respiratory symptoms for planes that do or do not recirculate air. The alternative hypothesis (\(H_a\)) claims there is a significant difference. Mathematically, \(H_0: p_1 = p_2\) and \(H_a: p_1 \neq p_2\) where \(p_1\) and \(p_2\) are the proportions of passengers with post-flight respiratory symptoms who flew on non-recirculating and recirculating planes respectively.

Calculate the Test Statistic

Let's denote the number of passengers who exhibited post-flight symptoms in non-recirculating and recirculating planes as \(x_1\) and \(x_2\), and the total number of passengers in both plane types as \(n_1\) and \(n_2\). Hence, \(x_1=108\), \(x_2=111\), \(n_1=517\), and \(n_2=583\). We can calculate \(p_1\), \(p_2\), and the pooled estimate of the proportion (\(p\)) as follows: \(p_1 = x_1/n_1\), \(p_2 = x_2/n_2\), \(p = (x_1 + x_2) / (n_1 + n_2)\). Then, we calculate the test statistic (z) with the formula \(z = (p_1 - p_2) / \sqrt{p(1-p)(1/n_1 + 1/n_2)}\).

Find the Critical Value and Make Decision

The critical value for a two-tailed test at \(\alpha = .05\) is approximately \(\pm 1.96\), from the standard normal distribution table. If our calculated z-value falls beyond this critical value, we reject the null hypothesis. If not, we fail to reject the null hypothesis.

Conclusion

Based on the decision from Step 3, we draw our conclusion. If we reject the null hypothesis, then there is sufficient evidence to conclude that the proportion of passengers with post-flight respiratory symptoms differs for planes that do and do not recirculate air. If we fail to reject the null hypothesis, then there is not sufficient evidence to support the claim.