91Ó°ÊÓ

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 mean passenger plus carry-on weights. They found an mean summer weight of 183 pounds and a winter mean of 190 pounds. Suppose that these estimates were based on random samples 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

Constructing a Confidence Interval for Summer

Step begins with plugging in the given values into the confidence interval formula. For summer weights, the mean, sample size and standard deviation are 183 pounds, 100, and 20 pounds respectively and for a 95% confidence interval, the Z value is 1.96, which can be found by looking up the standard normal distribution. So, the calculation would be: \(183 \pm 1.96 \frac{20}{\sqrt{100}}\) which results in: \(183 \pm 3.92\). Hence, the 95% confidence interval for the summer weights would be (179.08, 186.92) pounds.

02

Constructing a Confidence Interval for Winter

Repeat the same process for winter weights. The given mean, sample size and standard deviation for the winter weights are 190 pounds, 100 and 23 pounds respectively. The calculation would be: \(190 \pm 1.96 \frac{23}{\sqrt{100}}\) which results in: \(190 \pm 4.508\). Hence, the 95% confidence interval for the winter weights would be (185.492, 194.508) pounds.

03

Comparing the Confidence Intervals with FAA Recommendations

Now to comment on the FAA's recommendations in light of the obtained confidence intervals. The FAA's recommendation for summer is 190 pounds, which is clearly not in the confidence interval of summer, (179.08, 186.92) pounds. Therefore, the summer FAA recommendation may be too high. On the contrary, FAA's winter recommendation of 195 pounds is slightly above our winter confidence interval of (185.492, 194.508) pounds, suggesting that this recommendation is acceptable with our result.

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Key Concepts

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

Mean Estimation

Mean estimation is a central concept in statistics used to find the average value of a dataset. In the context of the problem above, the mean represents the average weight of passengers, including carry-on luggage, during summer and winter.
For instance, Frontier Airlines determined a summer mean of 183 pounds and a winter mean of 190 pounds, based on samples from 100 passengers each season. To estimate this mean accurately, collecting data from a random sample is crucial. This ensures that the calculated mean truly reflects the whole population, minimizing bias.
In practical scenarios, like the one illustrated, mean estimation helps airlines align their operations with realistic weight expectations, ensuring safety and regulatory compliance.

• Real sample of passengers: Ensures reliable estimates.
• Random selection: Speeds accuracy for broader populations.
• Effective mean estimation: Prevents bias and misinformation.

Sample Standard Deviation

The sample standard deviation measures the amount of variation or dispersion in a set of values. It tells us how much the weights of passengers vary around the mean weight. In our example, the summer and winter sample standard deviations are 20 pounds and 23 pounds, respectively.
A lower standard deviation indicates that values tend to be close to the mean, whereas a higher standard deviation shows a wider spread of values. In terms of airline passengers, understanding this spread is vital for safety and handling variations in passenger weights across different seasons.
Calculating the standard deviation before constructing a confidence interval is the first step. This helps assess the reliability of the mean estimate, as low deviation implies better accuracy.

• Summer deviation: 20 pounds.
• Winter deviation: 23 pounds.
• Insight on deviations: Gauges accuracy and reliability.

Normal Distribution

A normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. When working with confidence intervals, assuming a normal distribution helps in making reliable estimates.
In our problem, since the sample size is large (100 passengers), the Central Limit Theorem tells us that the sample mean can be treated as normally distributed. This justifies the use of the normal distribution in calculating the confidence intervals.
Moreover, using the standard normal distribution, the Z value of 1.96 is employed in calculating a 95% confidence interval. This Z value ensures that the interval includes the true mean weight of passengers with a high degree of certainty.

• Symmetry about the mean: Characteristic of normal distribution.
• Central Limit Theorem: Allows normal approximation for sample means.
• Z value: Ensures high confidence in mean estimation.