Question

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 \(180 \mathrm{lb}\) as a typical passenger weight (including carry-on luggage) in warm months and \(185 \mathrm{lb}\) 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 \(183 \mathrm{lb}\) and a winter average of \(190 \mathrm{lb} .\) Suppose that each of these estimates was based on a random sample of 100 passengers and that the sample standard deviations were 20 lb for the summer weights and \(23 \mathrm{lb}\) 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 \(190 \mathrm{lb}\) for summer and \(195 \mathrm{lb}\) for winter. Comment on these recommendations in light of the confidence interval estimates from Parts (a) and (b).

Short answer

The 95% confidence interval for the mean summer weight is approximately 183 \pm 3.92 lb, and for winter it's approximately 190 \pm 4.52 lb. It seems that the new FAA recommendations for passenger weights are reasonably aligned with these intervals, providing a level of validation for the new guidelines.

Step-by-step solution

Calculate Confidence Interval for Summer

First, let's construct a 95% confidence interval for the mean of the summer weight data. The formula for a confidence interval is \[X \pm Z* (S/ \sqrt{n} )\] where X is the sample mean, Z is the Z-score corresponding to the required confidence level (for 95% confidence, Z = 1.96), S is the sample standard deviation, and n is the sample size. Substituting the given values (X=183, S=20, n=100), we get \[183 \pm (1.96* (20/ \sqrt{100} ))\].

Calculate Confidence Interval for Winter

Next, let's construct a 95% confidence interval for the mean of the winter weight data. Using the same formula, we get \[190 \pm (1.96* (23/ \sqrt{100} ))\] using the given values (X=190, S=23, n=100).

Interpret the Confidence Intervals

Confidence intervals can be interpreted as the range within which we are 95% confident that the population parameter (in this case, passenger weight including carry-on luggage) lies.

Comment on the FAA Recommendations

The new FAA recommendations for passenger weights will be compared with the confidence intervals previously calculated for both the summer and winter weights. If the recommended weights lie within the confidence intervals, then they are likely valid, but if they lie outside, they may not be suitable.

Key concepts

Statistics Education

Understanding statistics is crucial for making informed decisions based on data. An excellent example of practical statistics at work is analyzing weight guidelines for airline passengers. Through analyzing samples, we gain insights into what the average passenger might weigh, which directly impacts safety standards. In this type of analysis, calculating confidence intervals is a core skill. It allows us to estimate with a certain confidence level, like 95%, where the true average weight lies based on a sample. Education in statistics equips students with the ability to analyze, interpret, and draw conclusions from data. Whether dealing with FAA guidelines or other real-world scenarios, statistics provide a robust framework for decision-making.

When students learn about confidence intervals, they discover the importance of sample size, variability within the data, denoted as sample standard deviation, and the concept of a Z-score, which is the number of standard deviations from the mean a data point is. These elements are foundational to not only understand FAA passenger weight guidelines but also many other practical applications in various fields.

FAA Passenger Weight Guidelines

The Federal Aviation Administration (FAA) is responsible for ensuring that airlines maintain high safety standards, which includes accurately estimating the weight of passengers and carry-on luggage. In 2003, FAA updated its passenger weight guidelines to reflect more accurate information, accounting for an increase in average weight over time. These weight estimates are fundamental in making calculations for fuel requirements, determining the number of passengers allowed on a plane, and ensuring the overall safety and stability of aircraft.

Implementing accurate guidelines requires thorough data analysis. Airlines may conduct studies using random samples of their passengers to estimate average weights. Studies like the one conducted by Frontier Airlines allow the FAA to evaluate the appropriateness of their recommended weights, which is why a statistical approach is necessary to ensure safety without overestimating and causing inefficiencies or underestimating and risking safety.

Sample Standard Deviation

Sample standard deviation is a measure used to quantify the amount of variability or dispersion in a set of data values. In simpler terms, it tells us how spread out the numbers in a sample are from the average (mean) value. A low standard deviation indicates that the data points tend to be close to the mean, whereas a high standard deviation indicates that the data points are spread out over a wider range of values.

In the exercise with FAA guidelines, knowing the sample standard deviation is crucial as it influences the width of the confidence interval. The larger the standard deviation, the wider the confidence interval, indicating more variability in passenger weights. Understanding this concept helps in precisely estimating how much variation to expect when setting weight guidelines for passengers. It is a crucial parameter in the confidence interval formula and significantly impacts the results and subsequent recommendations.

Z-score

The Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is identical to the mean score. A Z-score of 1.0 would signify that a value is one standard deviation from the mean; Z-scores can be positive or negative, indicating whether the value is above or below the mean, respectively.

In the context of constructing confidence intervals for the FAA passenger weight guidelines, the Z-score is used to determine the number of standard deviations an estimate should be from the sample mean to ensure a specified level of confidence. For a 95% confidence interval, the Z-score is 1.96, meaning that the true mean is expected to fall within 1.96 standard deviations of the sample mean 95% of the time. This Z-score helps us create a range where we believe the actual average weight of all passengers will fall, thus showing how Z-scores are essential in making predictions and decisions based on sample data.