Final answer:
To find the probability that a male passenger can fit through the doorway without bending, calculate the z-score and use a standard normal distribution table or calculator to find the probability of the z-score. In this case, the probability is approximately 99.38%.
Step-by-step explanation:
To find the probability that a randomly selected male passenger can fit through the doorway without bending, we need to find the z-score corresponding to the height of the doorway. The z-score is calculated by subtracting the population mean height from the doorway height and dividing by the standard deviation. Then, we use a standard normal distribution table or calculator to find the probability that a z-score is less than or equal to the calculated z-score.
In this case, the doorway height is 76 inches, the population mean height is 69 inches, and the standard deviation is 2.8 inches. The z-score is (76 - 69) / 2.8 = 2.5. Using a standard normal distribution table or calculator, we find that the probability of a z-score less than or equal to 2.5 is approximately 0.9938. Therefore, the probability that a randomly selected male passenger can fit through the doorway without bending is approximately 0.9938 or 99.38%.