Final answer:
To find the probability that a male passenger can fit through the doorway without bending, calculate the z-score for a height of 72 inches using the given mean and standard deviation. For the probability that the mean height of 150 men is less than 72 inches, calculate the standard error of the mean and the corresponding z-score. The result from part (b) is more relevant to engineers designing the 757.
Step-by-step explanation:
In order to find the probability that a male passenger can fit through the doorway without bending, we need to calculate the z-score for a height of 72 inches using the given mean and standard deviation. The formula to calculate the z-score is z = (x - mean) / standard deviation. Plugging in the values, we get z = (72 - 69) / 28 = 0.107. Next, we can use a z-table or a calculator to find the probability that a z-score is less than or equal to 0.107, which is approximately 0.541. Therefore, the probability that a male passenger can fit through the doorway without bending is approximately 0.541.
To find the probability that the mean height of 150 men is less than 72 inches, we first need to calculate the standard error of the mean (SEM) using the formula SEM = standard deviation / square root of sample size. Plugging in the values, we get SEM = 28 / sqrt(150) = 2.289. Next, we calculate the z-score for a mean height of 72 inches using the SEM, which is z = (72 - 69) / 2.289 = 1.310. Finally, we can use a z-table or a calculator to find the probability that a z-score is less than 1.310, which is approximately 0.905. Therefore, the probability that the mean height of 150 men is less than 72 inches is approximately 0.905.