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Passengers died when a water taxi sank in Baltimore's Inner Harbor. Men are typically heavier than women and children, so when loading a water taxi, assume a worst-case scenario in which all passengers are men. Assume that weights of men are normally distributed with a mean of \(189 \mathrm{lb}\) and a standard deviation of \(39 \mathrm{lb}\) (based on Data Set 1 "Body Data" in Appendix B). The water taxi that sank had a stated capacity of 25 passengers, and the boat was rated for a load limit of \(3500 \mathrm{lb}\). a. Given that the water taxi that sank was rated for a load limit of \(3500 \mathrm{lb}\), what is the maximum mean weight of the passengers if the boat is filled to the stated capacity of 25 passengers? b. If the water taxi is filled with 25 randomly selected men, what is the probability that thein mean weight exceeds the value from part (a)? c. After the water taxi sank, the weight assumptions were revised so that the new capacity became 20 passengers. If the water taxi is filled with 20 randomly selected men, what is the probability that their mean weight exceeds \(175 \mathrm{lb}\), which is the maximum mean weight that does not cause the total load to exceed \(3500 \mathrm{lb}\) ? d. Is the new capacity of 20 passengers safe?

Step by step solution

01

Maximum Mean Weight Calculation

First, calculate the maximum mean weight of passengers that the boat can handle when at full capacity (25 passengers). The load limit of the boat is given as 3500 lb: \[\text{Maximum Mean Weight} = \frac{\text{Total Load Limit}}{\text{Number of Passengers}} = \frac{3500}{25} = 140 \text{ lb} \]

02

Mean Weight Exceedance Probability Calculation (25 passengers)

Next, calculate the probability that the mean weight of 25 randomly selected men exceeds the maximum mean weight derived in Step 1 (140 lb). Use the given mean ( 189 lb ) and standard deviation ( 39 lb ) to find the z-score: \[ z = \frac{\bar{x} - \text{mean}}{\frac{\text{SD}}{\text{sqrt}(n)}} = \frac{140 - 189}{\frac{39}{\text{sqrt}(25)}} = \frac{140 - 189}{7.8} = -6.28 \]Then, consult the z-table or a calculator to find the probability corresponding to z = -6.28. Since this value is extremely small, the probability is effectively 0.

03

Mean Weight Exceedance Probability Calculation (20 passengers)

Calculate the probability that the mean weight of 20 randomly selected men exceeds 175 lb, which is the new maximum mean weight after revising the capacity. Use the given mean (189 lb) and standard deviation (39 lb) to find the z-score: \[ z = \frac{\bar{x} - \text{mean}}{\frac{\text{SD}}{\text{sqrt}(n)}} = \frac{175 - 189}{\frac{39}{\text{sqrt}(20)}} = \frac{175 - 189}{8.72} = -1.60 \]Consult the z-table or a calculator to find the probability corresponding to z = -1.60. The cumulative probability for z = -1.60 is about 0.0548. The probability of exceeding 175 lb is then: \[ P(\bar{x} > 175) = 1 - 0.0548 = 0.9452 \]

04

Safety Analysis of the New Capacity

Analyze if the new capacity of 20 passengers is safer by comparing each scenario. In this case, the revised assumption puts the probability of the load exceeding 3500 lb at approximately 5.52% when the capacity is at 20 passengers. Given this probability, the new capacity of 20 passengers is safer compared to 25 passengers, where the probability of exceeding the load limit was practically 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

normal distribution

In statistics, the normal distribution is one of the most important probability distributions. Also known as the bell curve, it shows that data near the mean are more frequent in occurrence than data far from the mean. For example, in the weight of men, most individuals will weigh close to the average weight, while fewer people will be much lighter or heavier. This concept is crucial when assessing the weight load for safety analysis in scenarios like the water taxi incident.

probability calculation

Probability calculations help us understand the likelihood of a certain event happening. In the context of the water taxi example, we calculate how likely it is for the mean weight of passengers to exceed a certain value. This involves the use of z-scores and the standard normal distribution. The calculations reveal the risks associated with different passenger capacities, contributing to safer load limit determinations.

z-score

A z-score indicates how many standard deviations an element is from the mean. It's a key tool in assessing probabilities. In our exercise, we calculated z-scores to see how far the mean weight of passengers deviates from the defined safety limits. For instance, with 25 passengers, the z-score for a mean weight of 140 lb was extremely low, showing a negligible probability of exceeding the load limit. Contrast this with 20 passengers and a z-score of -1.60 for 175 lb, yielding a non-negligible probability of about 5.52%.

maximum mean weight

The maximum mean weight of passengers is a critical factor in load limit analysis. It鈥檚 calculated by dividing the total load limit by the number of passengers. For the sinking water taxi, at a capacity of 25 passengers, each should not exceed a mean weight of 140 lb to stay within the 3500 lb limit. This calculation helps in determining safe operational limits by ensuring that the total load doesn't exceed the vessel's capacity.

load limit analysis

Load limit analysis involves examining how different passenger capacities affect the load limits for a vehicle or vessel. In this exercise, the original load limit was set for 25 passengers, but after the sinking, a new safer capacity was determined to be 20 passengers. This analysis considers probabilities, maximum mean weights, and standard deviations to ensure safety guidelines are not breached, thereby preventing accidents and promoting safer operational standards.