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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.

Step by step solution

01

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.

02

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 \(z = (p_1 - p_2) / \sqrt{p(1-p)(1/n_1 + 1/n_2)}\).

03

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.

04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportion Test

A proportion test is a statistical method used to determine if there is a significant difference between the proportions of two groups. In the context of our exercise, we're interested in comparing the proportion of passengers who experience post-flight respiratory symptoms on planes that recirculate air versus those that do not. The goal is to assess whether these proportions are significantly different from each other.

• We start by calculating the proportions for each group. For non-recirculating planes, we have \(p_1 = \frac{108}{517}\), and for recirculating planes, \(p_2 = \frac{111}{583}\).
• A pooled proportion \(p\) is also computed, which combines the data from both groups: \(p = \frac{108 + 111}{517 + 583}\).

This test is suitable when we suspect a difference but have no reason to believe one group has a greater proportion than the other.

Null Hypothesis

The null hypothesis is a fundamental concept in hypothesis testing. It represents a statement of no effect or no difference. In this exercise, the null hypothesis, denoted as \(H_0\), claims there is no difference in the proportions of passengers with symptoms between the two types of planes.Mathematically, it is stated as:

• \(H_0: p_1 = p_2\)

This implies that any observed difference in proportions is due to random sampling variability.
Notably, we test the null hypothesis against the alternative hypothesis, which suggests a significant difference exists.Understanding the role of the null hypothesis helps clarify whether the observed data supports a new claim or maintains the status quo.

Test Statistic Calculation

Calculating the test statistic allows us to quantify the difference between sample proportions. The test statistic in a proportion test is typically represented by a Z-score.To find the Z-score, follow these steps:

• Calculate the study proportions \(p_1\) and \(p_2\) for each group.
• Compute the pooled proportion \(p\).
• Use the formula:
  \[ z = \frac{(p_1 - p_2)}{\sqrt{p(1-p)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} \]

This Z-score tells us how many standard deviations away our observed difference is from zero under the null hypothesis.
If this value is sufficiently large (in either direction), it indicates the difference is statistically significant.

Critical Value

The critical value helps us decide whether to reject the null hypothesis. It is a threshold determined by the significance level, \(\alpha\), set for the test.For a two-tailed test with \(\alpha = 0.05\), the critical values are approximately \(\pm 1.96\). These values are derived from the standard normal distribution:

• If the calculated Z-score falls beyond these critical values, it indicates a significant difference, leading us to reject the null hypothesis.
• If the Z-score is within these limits, we fail to reject the null hypothesis, suggesting there isn't enough evidence to claim a significant difference.

The use of critical values provides a clear decision rule, ensuring consistent and objective conclusions based on statistical evidence.