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 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).

Short answer

The 95% confidence interval for the summer weights is approximately (179.08, 186.92) pounds, and for the winter weights is approximately (185.492, 194.508) pounds. The FAA's recommendation for summer is too high according to our results, but their winter recommendation seems to be acceptable according to our estimation.

Step-by-step solution

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.

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.

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.