When a water taxi sank in Baltimore's Inner Harbor, an...

When a water taxi sank in Baltimore's Inner Harbor, an investigation revealed that the safe passenger load for the water taxi was 3500 lb. It was also noted that the mean weight of a passenger was assumed to be 140 lb. Assume a “worstcase” scenario in which all of the passengers are adult men. (This could easily occur in a city that hosts conventions in which people of the same gender often travel in groups.) Assume that weights of men are normally distributed with a mean of 182.9 lb and a standard deviation of 40.8 lb (based on Data Set 1 in Appendix B). a. If one man is randomly selected, find the probability that he weighs less than 174 lb (the new value suggested by the National Transportation and Safety Board). b. With a load limit of 3500 lb, how many male passengers are allowed if we assume a mean weight of 140 lb? c. With a load limit of 3500 lb, how many male passengers are allowed if we use a new mean weight of 182.9 lb? d. Why is it necessary to periodically review and revise the number of passengers that are allowed to board?

When a water taxi sank in Baltimore's Inner Harbor, an investigation revealed that the safe passenger load for the water taxi was 3500 lb. It was also noted that the mean weight of a passenger was assumed to be 140 lb. Assume a “worstcase” scenario in which all of the passengers are adult men. (This could easily occur in a city that hosts conventions in which people of the same gender often travel in groups.) Assume that weights of men are normally distributed with a mean of 182.9 lb and a standard deviation of 40.8 lb (based on Data Set 1 in Appendix B).
a. If one man is randomly selected, find the probability that he weighs less than 174 lb (the new value suggested by the National Transportation and Safety Board).
b. With a load limit of 3500 lb, how many male passengers are allowed if we assume a mean weight of 140 lb?
c. With a load limit of 3500 lb, how many male passengers are allowed if we use a new mean weight of 182.9 lb?
d. Why is it necessary to periodically review and revise the number of passengers that are allowed to board?

Key Concepts

Normal Distribution

The normal distribution is a probability distribution that is symmetric around the mean and characterized by its bell-shaped curve. It is commonly used in statistics to model natural phenomena such as body weights, where most observations cluster around the mean with a predictable spread defined by the standard deviation.

Z-Score Standardization

Z-score standardization is the process of converting a normally distributed variable into a standard normal variable with a mean of 0 and a standard deviation of 1. This method simplifies the process of calculating probabilities by allowing comparisons against a universal distribution table, regardless of the original units of measurement.

Probability Calculation Using the Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) for a normal distribution represents the probability that a random variable is less than or equal to a specified value. In practical scenarios, the CDF is used to determine the likelihood of an event occurring, such as a person's weight falling below a certain threshold.

Load Capacity Calculation

Load capacity calculation involves using statistical averages—such as the mean weight—to determine the maximum number of individuals that can be safely supported by a given weight limit. This ensures that safety regulations are met by accounting for the expected load based on average conditions.

Risk Management and Periodic Review

Risk management in transportation safety includes regularly reviewing and updating safety standards based on the most recent and relevant empirical data. This is critical because population characteristics, such as average weights, can change over time, and outdated assumptions may lead to unsafe practices if not promptly revised.

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