91Ó°ÊÓ

Water Taxi Safety 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 lb and a standard deviation of 39 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 lb. a. Given that the water taxi that sank was rated for a load limit of 3500 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 their 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 lb, which is the maximum mean weight that does not cause the total load to exceed 3500 lb? d. Is the new capacity of 20 passengers safe?

Step by step solution

01

- Compute Maximum Mean Weight for 25 Passengers (Part a)

Calculate the maximum mean weight of passengers if the boat is filled to its stated capacity of 25 passengers with a load limit of 3500 lb. Let \( \bar{w} \) be the mean weight per passenger: \[ \bar{w} \times 25 = 3500 \text{ lb} \] \[ \bar{w} = \frac{3500 \text{ lb}}{25} = 140 \text{ lb} \]So, the maximum mean weight of the passengers must be 140 lb for 25 passengers to not exceed the load limit.

02

- Define and Calculate the Probability (Part b)

Given that mean weight \( \bar{w} \) of men is 189 lb and standard deviation \( \text{SD} \) is 39 lb, and assuming the weights of men are normally distributed, calculate the probability that the mean weight of 25 men exceeds 140 lb.To find this, standardize the value to a z-score: \[ z = \frac{\bar{X} - \text{mean}}{\frac{\text{SD}}{\text{sqrt}(n)}} \] \[ z = \frac{140 - 189}{\frac{39}{\text{sqrt}(25)}} = \frac{-49}{7.8} \ z \thickapprox -6.28 \] Use standard normal distribution tables to determine the probability. A z-score of -6.28 corresponds to an extremely low probability, practically 0.

03

- Calculate the Probability for 20 Passengers (Part c)

For 20 passengers, compute the probability that their mean weight exceeds 175 lb, which is the new capacity weight. Use the mean of 189 lb, standard deviation of 39 lb, and n=20. Calculate the z-score: \[ z = \frac{\bar{X} - \text{mean}}{\frac{\text{SD}}{\text{sqrt}(n)}} \] \[ z = \frac{175 - 189}{\frac{39}{\text{sqrt}(20)}} = \frac{-14}{8.72} \ z \thickapprox -1.61 \] Using standard normal distribution tables, find the probability for \( z = -1.61 \), which is about 0.0537 or 5.37%.

04

- Evaluate Safety of New Passenger Capacity (Part d)

Since the probability from part (c) that the mean weight of 20 passengers will exceed 175 lb is approximately 5.37%, which is relatively low, the new capacity of 20 passengers could be considered relatively safe.

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Key Concepts

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

Probability Theory

Probability theory is a branch of mathematics focused on analyzing random phenomena. In the context of this problem, we are calculating the likelihood that the mean weight of passengers on a water taxi exceeds certain values. This involves using probabilities to understand and predict events.

• We used the given mean and standard deviation to find probabilities.
• We calculated z-scores to relate our problem to the standard normal distribution.

Probability theory provides the foundation for these calculations.

Normal Distribution

A normal distribution is a symmetrical, bell-shaped curve that represents the distribution of many types of data. In this scenario, we assume that men's weights follow a normal distribution with a mean of 189 lb and a standard deviation of 39 lb.

The properties of normal distributions helped us:

• Determine probabilities for given values of passenger weights.
• Translate weights into z-scores, making it easier to find probabilities using standard normal distribution tables.

The normal distribution is crucial in understanding the probabilities of events in our analysis.

Statistical Analysis

Statistical analysis involves collecting, analyzing, and interpreting data. In our exercise, we performed statistical analysis by using known data (mean and standard deviation of men's weights) to make predictions.

• We calculated the maximum mean weight that each passenger can have without exceeding the boat's load limit.
• We determined the probabilities of exceeding certain weights by using statistical tools like the z-score.

This analysis helps ensure the safety of water taxi operations by providing data-driven insights.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. Here, we used the standard deviation of men's weights (39 lb) to calculate z-scores.

These z-scores tell us how many standard deviations a weight value is from the mean.

• A high standard deviation indicates more spread out weights.
• A lower standard deviation indicates weights are closer to the mean.

In our problem, the z-score calculations allowed us to use standard normal distribution tables to find probabilities easily.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions based on data. In our exercise, we implicitly conducted hypothesis testing to decide if the new passenger capacity (20 men) is safe.

• We tested the hypothesis that the mean weight of 20 passengers won't exceed 175 lb.
• The probability of exceeding 175 lb (5.37%) was relatively low, helping us conclude it's likely safe.

Hypothesis testing helps us make informed decisions based on statistical evidence.