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 $3500 \mathrm{lb}$. It was also noted that the mean weight of a passenger was assumed to be $140 \mathrm{lb}$. 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 $188.6 \mathrm{lb}$ and a standard deviation of $38.9 \mathrm{lb}$ (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 $174 \mathrm{lb}$ (the new value suggested by the National Transportation and Safety Board). b. With a load limit of $3500 \mathrm{lb}$, how many male passengers are allowed if we assume a mean weight of $140 \mathrm{lb}$ ? c. With a load limit of $3500 \mathrm{lb}$, how many male passengers are allowed if we assume the updated mean weight of $188.6 \mathrm{lb}$ ? 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 $3500 \mathrm{lb}$. It was also noted that the mean weight of a passenger was assumed to be $140 \mathrm{lb}$. 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 $188.6 \mathrm{lb}$ and a standard deviation of $38.9 \mathrm{lb}$ (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 $174 \mathrm{lb}$ (the new value suggested by the National Transportation and Safety Board).
b. With a load limit of $3500 \mathrm{lb}$, how many male passengers are allowed if we assume a mean weight of $140 \mathrm{lb}$ ?
c. With a load limit of $3500 \mathrm{lb}$, how many male passengers are allowed if we assume the updated mean weight of $188.6 \mathrm{lb}$ ?
d. Why is it necessary to periodically review and revise the number of passengers that are allowed to board?

Key Concepts

Probability Calculation

Probability calculation involves determining the likelihood of an event occurring within a given distribution. In the context of normal distribution, this is typically done by converting raw scores to z-scores and then using standard normal distribution tables or calculators to find the probability that a randomly selected value falls below, above, or between certain values.

Periodic Reevaluation of Safety Standards

Periodic reevaluation of safety standards is necessary because underlying population characteristics, such as weight distributions, can change over time due to shifts in demographics or health trends. Regular reviews ensure that the safety assessments, load limits, and regulatory guidelines remain current and effective in mitigating risks, thereby protecting public safety and ensuring operational reliability.

Statistical Assumptions

Statistical assumptions are the underlying premises about data, such as the shape of the distribution (e.g., normality), the values for mean and standard deviation, and the applicability of these parameters to the entire population. These assumptions are critical in designing safety protocols and making decisions, yet they can be a source of error if the actual data differ significantly from the assumed model.

Z-Score

The z-score is a statistical measure that describes a value's position in relation to the mean of a group of values, measured in terms of standard deviations. It allows for the standardization of different data points from a normal distribution, making it possible to compute probabilities and compare scores across different normal distributions.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, symmetry about the mean, and parameters (mean and standard deviation). This distribution is widely used in statistics to model natural phenomena including human weights, measurement errors, and many other types of data where variability is expected around an average value.

Load Capacity Calculation

Load capacity calculation refers to the process of determining the maximum allowable weight based on a predetermined safety limit. This involves dividing the total allowable weight by an assumed mean passenger weight to estimate the maximum number of admissible passengers. Relying solely on an assumed average, however, may oversimplify the risk in situations where the actual weight distribution deviates from the assumed norm.

Transcript

00:01 All right, so we have a water taxi sink in baltimore's inner harbor, and we're talking about safe passenger loads for these water taxis.

00:08 And we know that 3 ,500 pounds is the limit.

00:16 And i noted that the mean weight of a passenger was assumed 340 pounds.

00:22 Assume the worst case scenario in which all passengers are adult men.

00:27 We want to know a couple things here first.

00:30 If one man is randomly selected, what's the probability that man lays less than 174 pounds, which is the new value suggested by the national transportation safety board based on the seemingly outdated information of 140 pounds? so what we're looking for here actually is the probability that x is less than 174.

00:54 And because these weights are normally distributed, that's the same as the probability that z is less than x value, which is 174, minus the mean of all men passengers, 108 .6, divided by their standard deviation 38 .9, which gives us a z score of negative .375.

01:21 And we can get the actual probability from either a table or a calculator.

01:26 However you get it, we get a probability of 0 .354.

01:31 So the probability that a single passenger weighs less than a 174 pounds is about 35%.

01:42 Next, we want to know how many mail passengers could the boat hold with a load limit of 3 ,400, assuming they weigh 140 pounds.

01:54 So you'd the old method.

01:57 Because this is the mean weight of all passengers, they're going to be equal to the, total weight, which is 3 ,500, divided by the number of passengers.

02:25 We know the mean weight we said was 140.

02:30 And we'll call the number of passengers x.

02:33 So if we cross multiply here in software x, we get 140x...

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