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Because of safety considerations, in May 2003 the Federal Aviation Administration (FAA) changed its guidelines for how small commuter airlines must estimate passenger weights. Under the old rule, airlines used 180 pounds as a typical passenger weight (including carry-on luggage) in warm months and 185 pounds as a typical weight in cold months. The Alaska Journal of Commerce (May 25,2003\()\) reported that Frontier Airlines conducted a study to estimate average passenger plus carry-on weights. They found an average summer weight of 183 pounds and a winter average of 190 pounds. Suppose that each of these estimates was based on a random sample of 100 passengers and that the sample standard deviations were 20 pounds for the summer weights and 23 pounds for the winter weights. a. Construct and interpret a \(95 \%\) confidence interval for the mean summer weight (including carry-on luggage) of Frontier Airlines passengers. b. Construct and interpret a \(95 \%\) confidence interval for the mean winter weight (including carry-on luggage) of Frontier Airlines passengers. c. The new FAA recommendations are 190 pounds for summer and 195 pounds for winter. Comment on these recommendations in light of the confidence interval estimates from Parts (a) and (b).

Step by step solution

01

Formula for Confidence Interval

Firstly, let's remind ourselves of the formula we need to calculate a confidence interval. A confidence interval can be computed as follows using following formula:\n\[Confidence\ Interval = X̄ ± (Z_{\frac{α}{2}} * \frac{σ}{\sqrt{n}})\]\nIn this equation:\n- \(X̄\) represents the sample mean\n- \(Z_{\frac{α}{2}}\) is the z-score associated with the desired confidence level, 1.96 for a 95% confidence level\n- \(σ\) is standard deviation.\n- \(n\) is the size of the sample.

02

Calculate the Confidence Interval for Summer

Given that the summer mean, \(X̄_summer = 183\), standard deviation, \(σ_summer = 20\), and sample size for summer, \(n_summer = 100\), we substitute these values into our formula and calculate.\nTherefore, the confidence interval for summer weight will be\n\[183 ± (1.96 * \frac{20}{\sqrt{100}})\]

03

Calculate the Confidence Interval for Winter

Similarly, using the winter mean, \(X̄_winter = 190\), standard deviation, \(σ_winter = 23\), and sample size for winter, \(n_winter = 100\), we'll substitute these values into our formula and calculate.\nTherefore, the confidence interval for winter weight will be \n\[190 ± (1.96 * \frac{23}{\sqrt{100}})\]

04

Interpret the Results

After calculating the confidence intervals, we'll compare them to the FAA's new recommendations and interpret the results.

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Key Concepts

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

Descriptive Statistics

Descriptive statistics is a critical part of analyzing data and it provides summary statistics that describe the basic features of a dataset. When you hear terms like 'mean', 'median', 'mode', 'variance', and 'standard deviation', you are dealing with descriptive statistics. These tools allow researchers to present quantitative descriptions in a manageable form.

In the context of the FAA weight guidelines case study, the mean weights—183 pounds for summer and 190 pounds for winter—are examples of the use of descriptive statistics. The means tell us the average weight of the passengers during each season. In addition to the mean, the standard deviation is provided, which is a measure of the amount of variation or dispersion of a set of values. Standard deviations of 20 pounds for summer and 23 pounds for winter indicate the extent to which individual passenger weights differ from the mean weight for that season.

FAA Weight Guidelines

The FAA weight guidelines are critical for ensuring the safety and efficiency of air travel. Compliance with these guidelines helps to avoid overloading and can assist in the calculation of fuel requirements, which is based on the overall weight the aircraft will carry. Following the assessment of Frontier Airlines' average passenger weights, the FAA made recommendations that were higher than the previous estimates, adjusting the typical summer passenger weight from 180 to 190 pounds and the winter weight from 185 to 195 pounds.

It's essential for these guidelines to be updated periodically to reflect changes in population weight trends. Obesity rates, changes in fashion, and other factors can influence the average weight of passengers over time. The study done by Frontier Airlines represents an important piece of data used to inform these guidelines and ensure that they remain accurate and practical.

Statistical Inference

Statistical inference refers to the process of drawing conclusions from data that are subject to random variation. This includes inferential statistics methods such as confidence interval calculation, hypothesis testing, and regression analysis. Confidence intervals, in particular, are a way to estimate the value of a population parameter, such as the mean, based on sample data. They provide a range of values that with a certain degree of confidence contain the parameter of interest.

In our exercise, the confidence intervals allow us to infer—with 95% confidence—the true average weights of Frontier Airlines passengers in summer and winter. If we repeatedly took samples and built intervals in this way, 95% of the intervals would capture the true mean passenger weights. The FAA guidelines would ideally be compared against these intervals to determine if they are likely to be safe choices for average passenger weight estimates across all flights.