A boat capsized and sank in a lake. Based on an assumption of a mean weight of 142 lb , the boat was rated to carry 70 passengers (so the load limit was \( 9,940 \mathrm{lb} \). After the boat sank, the assumed mean w for similar boats was changed from 142 lb to 172 lb . Complete parts a and b below. Part 10 of 3 a. Assume that a similar boat is loaded with 70 passengers, and assume that the weights of people are normally distributed with a mean of 182.1 lb and a standard deviation of 38.13 lb of 16 points boat is overloaded because the 70 passengers have a mean weight greater than 142 lb . The probability is (Round to four decimal places as needed.)

To find the probability that 70 passengers on a similar boat (with a mean weight of 182.1 lb) would be considered overloaded, we perform a normal distribution calculation. 1. First, we calculate the standard error (SE) for the sample mean: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{38.13}{\sqrt{70}} \approx 4.54 \] 2. Next, we determine the Z-score for the mean weight of 142 lb: \[ Z = \frac{\bar{x} - \mu}{SE} = \frac{142 - 182.1}{4.54} \approx -8.83 \] 3. Finally, we look up this Z-score in a standard normal distribution table or use a calculator: The probability of a Z-score of -8.83 is essentially 0 (indicating that the mean weight of the passengers is likely much heavier than the limit). So, the probability that the boat is overloaded with 70 passengers who each weigh on average 182.1 lb is effectively 0.0000 when rounded to four decimal places. Now let’s explore how this situation connects to the broader themes of boat safety and relevant regulations. In the realm of maritime safety, weight limits are critical. The rating of a boat's capacity is typically based on standard weight assumptions, but these can vary significantly with increases in average passenger weight, not only affecting buoyancy but also stability. An overloaded vessel can capsize even in calm waters, leading to catastrophic accidents. Historically, boat safety regulations have evolved in response to sea tragedies. Notable incidents like the Titanic disaster prompted stricter guidelines on lifeboat capacity and passenger weight limits, instilling a culture of safety that emphasizes assessing real-world conditions including passenger demographics, weather, and load distribution before embarking on a journey.