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 ib It was also noted that the mean weight of a passenger was assumed to be 140 ib. Assume a "worst-case" scenario in which all of the passengers are adult men. Assume that weights of men are normally distributed with a mean of 188.6 Ib and a standard deviation of 38.9 Ib (based on Data Set 1 "Body Data" 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 ib, how many male passengers are allowed if we assume a mean weight of 140 lb? c. With a load limit of 3500 ib, how many male passengers are allowed if we assume the updated mean weight of 188.6 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 ib It was also noted that the mean weight of a passenger was assumed to be 140 ib. Assume a "worst-case" scenario in which all of the passengers are adult men. Assume that weights of men are normally distributed with a mean of 188.6 Ib and a standard deviation of 38.9 Ib (based on Data Set 1 "Body Data" 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 ib, how many male passengers are allowed if we assume a mean weight of 140 lb?
c. With a load limit of 3500 ib, how many male passengers are allowed if we assume the updated mean weight of 188.6 lb?
d. Why is it necessary to periodically review and revise the number of passengers that are allowed to board?

Key Concepts

Periodic Review and Revision of Safety Standards

Safety standards and regulations must be reviewed and updated periodically as new data, improved measurements, or changes in population characteristics emerge. This process ensures that assumptions used for planning, such as average weights or capacities, reflect current realities, thereby maintaining public safety and optimizing resource utilization.

Load Limit Determination Based on Statistical Assumptions

In engineering and safety planning, average values and distributions are used to estimate the capacity or load limits of structures and vehicles. This involves using measures of central tendency, such as the mean, along with variability measures like the standard deviation, to ensure that risk is managed even under worst-case scenarios.

Z-Score Calculation

A Z-score is a standardized measure that indicates how many standard deviations an element is from the mean of its distribution. It facilitates the comparison of data points from different normal distributions and allows for the use of probability tables to calculate the likelihood of observing values below or above a certain threshold.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean, where data near the mean are more frequent in occurrence than data far from the mean. It serves as a foundational model in statistics for naturally occurring phenomena and is critical for making probabilistic inferences about sample data in many fields, including safety engineering and quality control.

Probability Calculation Using the Normal Model

Calculating probabilities using the cumulative distribution function (CDF) of the normal distribution is a common technique in statistics, enabling the determination of the likelihood that a normally distributed variable will fall within a specific range. This is essential for risk assessments and decision making based on statistical evidence.

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