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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

Compute Summer Confidence Interval

The mean (sample estimate) for summer weight is 183 pounds, the standard deviation (s) is 20 pounds, and the sample size (n) is 100. Since we are asked to construct a 95% confidence interval, we need to identify the Z value for a 95% confidence interval from the Z-table, which is approximately 1.96. The formula for the confidence interval is: \[ \bar{X} \pm Z * \frac{s}{\sqrt{n}} \]. Plugging in the numbers we get: \[ 183 \pm 1.96 * \frac{20}{\sqrt{100}} \]

02

Compute Winter Confidence Interval

The mean (sample estimate) for winter weight is 190 pounds, the standard deviation (s) is 23 pounds, and the sample size (n) is 100. Using the same Z value from the Z table for a 95% confidence interval (1.96), the formula for the confidence interval is: \[ \bar{X} \pm Z * \frac{s}{\sqrt{n}} \]. Plugging in the numbers, we get: \[ 190 \pm 1.96 * \frac{23}{\sqrt{100}} \]

03

Interpret the Confidence Intervals

The calculated confidence intervals provide an estimated range for the mean weight of passengers in both summer and winter. The intervals tell us that we're 95% confident that the true average weight lies within these ranges.

04

Comment on FAA Recommendations

The FAA recommendations can be considered in light of the constructed confidence intervals. These should be compared with the calculated confidence intervals. If FAA's recommended weights do not fall within these intervals, it suggests that their estimations might not be accurate.

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