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

The short answer would be determined based on the p-value and the level of significance. If the p-value is less than .05, there is enough evidence to reject the null hypothesis and conclude that the proportion of passengers with post-flight respiratory symptoms differs for planes that do and do not recirculate air. If the p-value is greater than .05, there is not enough evidence to reject the null hypothesis, meaning we can't conclude that there is a difference in the proportions.

Step-by-step solution

State the hypotheses

The null hypothesis (\(H_0\)): The proportion of passengers with post-flight respiratory symptoms on planes that do and don't recirculate air is the same. The alternative hypothesis (\(H_1\)): The two proportions are not equal.

Calculate sample proportions

Calculate the proportion of passengers with post-flight respiratory symptoms for each group. For the non-recirculating air group, the proportion (\(p_1\)) is 108/517. For the recirculating air group, the proportion (\(p_2\)) is 111/583.

Calculate the standard error

Calculate the standard error using the formula \(SE = \sqrt{p(1 - p)(1/n_1 + 1/n_2)}\) where \(p\) is the pooled sample proportion, calculated as \((x_1 + x_2) / (n_1 + n_2)\) (x - number of successes in a sample, n - sample size).

Compute the test statistic

Calculate the test statistic using the formula \(Z = (p_1 - p_2) / SE\).

Calculate the p-value

The p-value is the probability of observing a sample statistic as extreme as the test statistic. Since the alternative hypothesis is not directional (it doesn't state which proportion is larger), use the two-tail approach for calculating the p-value. Refer to the standard normal distribution table to find the probability associated with the calculated Z.

Conclude the hypothesis test

If the p-value is less than the level of significance (\(\alpha = .05\)), reject the null hypothesis. If the p-value is greater than \(\alpha\), do not reject the null hypothesis. Based on this decision, conclude if there is enough evidence to say that the proportion of passengers with post-flight respiratory symptoms differs for planes that do and do not recirculate air.

Key concepts

Null Hypothesis

The null hypothesis, often symbolized as \(H_0\), is a statement that suggests there is no effect or no difference in a given context. In hypothesis testing, it's the claim that there's no change or that a certain parameter equals a specific value.
In the context of airplane cabin air recirculation, and its connection with post-flight respiratory symptoms, our null hypothesis posits that the proportion of travelers experiencing symptoms is the same, whether or not the aircraft recirculates air.
This hypothesis serves as the starting point for our analysis. It's the idea we aim to test against, using statistical methods, to determine if observed data like collected samples significantly differ, warranting a rejection of this hypothesis.

Alternative Hypothesis

The alternative hypothesis, represented as \(H_1\), suggests that there is an effect or a difference. It's what researchers hope to support through their analysis.
For the airline study, the alternative hypothesis proposes that the proportion of passengers with respiratory symptoms does vary between those on aircrafts with recirculated air versus those without. By identifying this hypothesis, researchers set out to find if the proportions differ, indicating that air recirculation could influence symptom incidence.
The alternative hypothesis is pivotal, as it gives direction to the test, either affirming a change or highlighting a difference when compared to the null hypothesis.

Standard Error

The standard error (SE) measures the accuracy with which a sample represents a population. It is calculated using sample proportions and provides insight into the variability of the data.
In the context of our exercise, we compute the standard error to assess how much the sample proportions of respiratory symptoms among passengers could differ due to random sampling.
The formula for the standard error in comparing two proportions is: \[ SE = \sqrt{p(1 - p)(\frac{1}{n_1} + \frac{1}{n_2})} \] where \( p \) is the pooled sample proportion, \( n_1 \) is the sample size for non-recirculating planes, and \( n_2 \) for recirculating planes.
This computation is crucial for determining if observed differences are statistically significant.

P-Value

The p-value is a probability measure that helps determine the significance of results obtained from a statistical hypothesis test.
A low p-value indicates that the observed data is highly unlikely under the assumption that the null hypothesis is true. Conversely, a high p-value suggests that observed data is plausible when the null hypothesis holds.
In our scenario, we conduct a two-tailed test because our alternative hypothesis only states that the proportions are not equal, without specifying which is greater. We then calculate the p-value using the standard normal distribution table and our test statistic \( Z \).
This value will tell us if the deviations we observe in our test are just due to random chance or if they suggest real differences between the groups.

Z-Test

The Z-test is a statistical method used to determine if there is a significant difference between sample and population means, or among two sample means. It's appropriate when data follows a normal distribution, and we know the population variance. In our case, we compare proportions.
The Z-test formula used to find the test statistic is: \[ Z = \frac{(p_1 - p_2)}{SE} \] where \( p_1 \) and \( p_2 \) are the respective sample proportions of passengers experiencing symptoms in the respective groups, and \( SE \) is the computed standard error.
Once calculated, this Z-value indicates how far our sample's observation deviates from the null hypothesis. We use this in conjunction with the p-value to decide whether to reject the null hypothesis, providing insights into our original research question.