Final answer:
The question entails using statistical methods to address engineering and business optimization in airline operations, including doorway height for most passengers and estimations for unoccupied seats. It involves calculations using z-scores and hypothesis testing.
Step-by-step explanation:
The question offered presents several problems related to the application of statistical analysis in the engineering and business management of airlines and aircraft design. These problems include determining the mean doorway height to accommodate most men without bending, estimating the mean number of unoccupied seats per flight, and assessing the value of R&D projects in relation to a private rate of return.
Answering a), to find the mean doorway height that allows 95 percent of men to enter without bending, one would use the z-score corresponding to the 95th percentile and apply it to the formula: mean height + (z-score * standard deviation). Since the population mean is 69 inches and the standard deviation is 2.8 inches, we would look up the z-score for 95% in a standard normal distribution table, which is approximately 1.645. Therefore, the required height is approximately 69 + (1.645 * 2.8) inches.
For b), the mean doorway height for the condition with a 0.95 probability would also use a z-score but would consider the mean of means distribution for 100 men. Because the standard deviation of the mean would be the population standard deviation divided by sqrt(n), the standard deviation would be smaller, thus requiring a different calculation.
Regarding the unoccupied seats problem, hypothesis testing would be employed to determine if the airline executive's belief about the average number of babies on flights holds true based on the sample data.