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Does the airline you choose affect when you'll arrive at your destination? The dataset DecemberFlights contains the difference between actual and scheduled arrival time from 1000 randomly sampled December flights for two of the major North American airlines, Delta Air Lines and United Air Lines. A negative difference indicates a flight arrived early. We are interested in testing whether the average difference between actual and scheduled arrival time is different between the two airlines. (a) Define any relevant parameter(s) and state the null and alternative hypotheses. (b) Find the sample mean of each group, and calculate the difference in sample means. (c) Use StatKey or other technology to create a randomization distribution and find the p-value. (d) At a significance level of \(\alpha=0.01\), what is the conclusion of the test? Interpret the conclusion in context.

Step by step solution

01

Define Parameters and State Hypotheses

This study’s parameters would be the actual differences in arrival times for the two airlines, denoted as µ_1 for Delta Air Lines and µ_2 for United Air Lines. The null hypothesis (H_0) is that the difference in the average actual vs. scheduled arrival times (µ_1-µ_2) is the same for both airlines. People often state the null hypothesis as µ_1-µ_2 = 0 (which implies the same thing). The alternative hypothesis (H_a) is that µ_1-µ_2 ≠ 0 meaning there is a difference between the two airlines.

02

Find Sample Means and their Difference

Calculate the sample mean difference in arrival times for each airline. Then subtract the Delta Air Lines mean from the United Air Lines mean to get the observed difference in sample means, denoted as d_obs.

03

Carry Out a Permutation Test

Use statistical software to perform a permutation test and generate a randomization distribution. This will simulate the distribution under the null hypothesis and allows the calculation of the p-value.

04

Find the p-value

By comparing the observed mean difference (d_obs) to the randomization distribution, the p-value can be computed. The p-value shows the probability of getting a mean difference as extreme as d_obs or more under the null hypothesis

05

Interpret and Make Conclusion

At a significance level of \(\alpha=0.01\), if the p-value is less than \(\alpha\), reject the null hypothesis. This suggests that there is a significant difference in the flight arrival times between the two airlines. If the p-value is greater than \(\alpha\), fail to reject the null hypothesis, which would mean there is no significant difference between the flight arrival times of the two airlines.

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Key Concepts

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

Null Hypothesis

In hypothesis testing, the null hypothesis is a statement that there is no effect or no difference. It is the assumption that any observed difference in a dataset is due to chance. In the context of the flight arrival times for Delta and United Airlines, the null hypothesis (\(H_0\)) asserts that the average difference in arrival times is the same for both airlines. In simpler terms, it means that both airlines arrive around the same time on average. We express this mathematically as \(\mu_1 - \mu_2 = 0\), where \(\mu_1\) and \(\mu_2\) are the mean arrival differences for Delta and United Airlines, respectively.

Alternative Hypothesis

The alternative hypothesis provides a statement that contradicts the null hypothesis. It suggests that there is a significant difference between groups being compared, indicating an effect exists. For our airline example, the alternative hypothesis (\(H_a\)) is that there is a significant difference between the average arrival times of Delta and United Airlines. This is expressed as \(\mu_1 - \mu_2 eq 0\). Under the alternative hypothesis, we suppose that factors other than random variation are responsible for any observed difference between the airlines.

Randomization Distribution

A randomization distribution is a crucial concept in hypothesis testing. It represents the possible outcomes of a test statistic (like mean difference) under the null hypothesis. For example, in a permutation test, the data is shuffled many times to simulate the process of random assignment. This creates a distribution of outcomes that might happen purely by chance if the null hypothesis were true. For the airline data, we would repeatedly sample the arrival time differences randomly between Delta and United Airlines. By examining the randomization distribution, we can see how unusual our observed sample difference is if no real difference exists between the airlines.

Significance Level

The significance level, denoted by \(\alpha\), is a threshold chosen by the researcher to decide whether to reject the null hypothesis. It is the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. Common significance levels are 0.05 or 0.01, with lower levels indicating stricter criteria for finding evidence against \(H_0\). In our scenario, \(\alpha = 0.01\) means we are using a strict criterion. We require strong evidence from the data to conclude there’s a difference in the airlines’ arrival times, ensuring our conclusion is less likely a result of random chance.

P-value

A p-value measures the strength of evidence against a null hypothesis. It tells us the probability of observing data as extreme, or more extreme, than what was actually observed, assuming the null hypothesis is true. When we perform our statistical test on the flight data, we calculate a p-value by comparing the observed difference (the mean arrival time difference) to the randomization distribution. If the p-value is less than the significance level (\(\alpha = 0.01\) in our case), we reject the null hypothesis. A small p-value suggests that the observed difference is unlikely to have occurred by random chance alone, supporting the idea that there is a true difference between Delta and United Airlines arrival times.