Question

Flying Home for the Holidays, On Time In Exercise 4.115 on page \(302,\) we compared the average difference between actual and scheduled arrival times for December flights on two major airlines: Delta and United. Suppose now that we are only interested in the proportion of flights arriving more than 30 minutes after the scheduled time. Of the 1,000 Delta flights, 67 arrived more than 30 minutes late, and of the 1,000 United flights, 160 arrived more than 30 minutes late. We are testing to see if this provides evidence to conclude that the proportion of flights that are over 30 minutes late is different between flying United or Delta. (a) State the null and alternative hypothesis. (b) What statistic will be recorded for each of the simulated samples to create the randomization distribution? What is the value of that statistic for the observed sample? (c) Use StatKey or other technology to create a randomization distribution. Estimate the p-value for the observed statistic found in part (b). (d) At a significance level of \(\alpha=0.01\), what is the conclusion of the test? Interpret in context. (e) Now assume we had only collected samples of size \(75,\) but got essentially the same proportions (5/75 late flights for Delta and \(12 / 75\) late flights for United). Repeating steps (b) through (d) on these smaller samples, do you come to the same conclusion?

Short answer

The answer depends on the estimated p-value. If the p-value is less than 0.01, we reject the null hypothesis and conclude that there is a significant difference in late arrival proportions between Delta and United. Otherwise, we do not reject the null hypothesis. The conclusion might change when repeating the process with samples of size 75.

Step-by-step solution

State the hypothesis

The null hypothesis (H0) is that the proportion of flights over 30 minutes late is the same for United and Delta. The alternative hypothesis (H1) is that the proportion is different for the two airlines.

Calculate the observed statistic

The statistic in this case is the difference in proportions. For Delta, the proportion is \( \frac{67}{1000} = 0.067 \). For United, the proportion is \( \frac{160}{1000} = 0.16 \). Hence, the observed difference is \( 0.16 - 0.067 = 0.093 \). This statistic will be calculated for each simulated sample.

Create a randomization distribution and estimate the p-value

By simulating a large number of samples, one can count the number of times the difference in proportions exceeds the observed statistic of 0.093. Divide this count by the total number of simulated samples to get the p-value. Note that the p-value is estimated using statistical software or tools like StatKey.

Interpret the results in regards to the alpha level

If the p-value is less than alpha (\( \alpha = 0.01 \)), reject the null hypothesis and conclude there is evidence of a difference in proportions. Otherwise, do not reject the null hypothesis, meaning the observed difference could be by random chance alone.

Repeat steps for samples of size 75

Assume that samples of size 75 were collected and proportions were \( \frac{5}{75} \) for Delta and \( \frac{12}{75} \) for United. Perform steps (b) to (d) again. If the conclusion changes because of the different sample size, it might imply that the sample size affects the power of the test.

Key concepts

Null and Alternative Hypothesis

Understanding the null and alternative hypotheses is crucial in hypothesis testing. These hypotheses are mutually exclusive statements about a population parameter, often involving population proportions or means.

In our exercise, the null hypothesis (\(H_0\)) posits that there is no difference in the proportion of flights arriving more than 30 minutes late between Delta and United Airlines. Mathematically, this can be expressed as \(P_{Delta} = P_{United}\). Conversely, the alternative hypothesis (\(H_1\)) suggests that there is indeed a difference, and it's formulated as \(P_{Delta} eq P_{United}\).

The goal of hypothesis testing is to determine which hypothesis the data supports, using a pre-determined significance level to judge whether the observed differences are statistically significant, or if they could have occurred by chance.

Proportions Comparison

When comparing proportions, such as the exercise's percentage of late flights for two airlines, we calculate the difference between the two sample proportions. This difference is pivotal in the hypothesis testing process. For the given data, the proportions of late flights for Delta and United are \(0.067\) and \(0.16\), respectively.

The observed statistic, the difference in these two proportions (\(0.16 - 0.067 = 0.093\)), serves as the basis for comparing what we would expect if the null hypothesis were true to what we actually observe. A significant difference implies that the airlines do not perform equally relative to the specified criterion, prompting further investigation into the statistical significance of this observed difference.

Randomization Distribution

A randomization distribution is a powerful tool in statistics that enables us to understand what kind of sample statistics we might observe if the null hypothesis were true. To create this, we simulate taking many samples from a combined population that assumes the null hypothesis is true, then calculate the statistic of interest for each one.

In our case, the statistic is the difference in late flight proportions. We use computer simulations to produce a distribution of this difference assuming no real difference between the airlines. This randomization distribution forms the backdrop against which we compare our observed statistic, allowing us to make inferences about the likelihood that our observed difference occurred under the null hypothesis.

P-Value Estimation

The p-value is essential in hypothesis testing; it quantifies the probability of observing a statistic as extreme as, or more extreme than, the one calculated from our sample data, assuming that the null hypothesis is true. To estimate the p-value, statistical software or tools like StatKey are typically employed.

Upon creating a randomization distribution, we estimate the p-value by determining the proportion of simulated differences that are as large as or larger than our observed difference of \(0.093\). A small p-value indicates that such an extreme result is unlikely under the null hypothesis, suggesting that the observed difference may be attributable to an actual effect rather than random variation.

Statistical Significance

The concept of statistical significance pertains to the decisiveness of a hypothesis test result. It determines whether the evidence in the sample data is strong enough to reject the null hypothesis. This is done by comparing the p-value to a pre-set alpha level (\(\alpha\)), which is the threshold for statistical significance.

In our exercise, the alpha level is set at \(0.01\), indicating a 1% risk of concluding that there is a difference in late flight proportions when there is none. If the p-value is smaller than this alpha level, we reject the null hypothesis, implying that there is statistically significant evidence of a difference. Conversely, if the p-value is higher, we do not reject the null hypothesis, suggesting that the observed difference could be just a result of sampling variability.