Questions: A water taxi carries passengers from harbor to another. Assume that weights of passengers are normally distributed with a mean of 182 lb and a standard deviation of 45 lb. The water taxi has a stated capacity of 25 passengers, and the water taxi was rated for a load limit of 3500 lb. Complete parts (a) through (d) below. a. Given that the water taxi was rated for a load limit of 3500 lb, what is the maximum mean weight of the passengers if the water taxi is filled to the stated capacity of 25 passengers? The maximum mean weight is 140 lb. (Type an integer or a decimal. Do not round.) b. If the water taxi is filled with 25 randomly selected passengers, what is the probability that their mean weight exceeds the value from part (a)? The probability is 1.0000. (Round to four decimal places as needed.) c. If the weight assumptions were revised so that the new capacity became 20 passengers and the water taxi is filled with 20 randomly selected passengers, 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? The probability is . (Round to four decimal places as needed.)

A water taxi carries passengers from harbor to another. Assume that weights of passengers are normally distributed with a mean of 182 lb and a standard deviation of 45 lb. The water taxi has a stated capacity of 25 passengers, and the water taxi was rated for a load limit of 3500 lb. Complete parts (a) through (d) below. a. Given that the water taxi was rated for a load limit of 3500 lb, what is the maximum mean weight of the passengers if the water taxi is filled to the stated capacity of 25 passengers? The maximum mean weight is 140 lb. (Type an integer or a decimal. Do not round.) b. If the water taxi is filled with 25 randomly selected passengers, what is the probability that their mean weight exceeds the value from part (a)? The probability is 1.0000. (Round to four decimal places as needed.) c. If the weight assumptions were revised so that the new capacity became 20 passengers and the water taxi is filled with 20 randomly selected passengers, 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? The probability is . (Round to four decimal places as needed.)
Transcript text: A water taxi carries passengers from harbor to another. Assume that weights of passengers are normally distributed with a mean of 182 lb and a standard deviation of 45 lb. The water taxi has a stated capacity of 25 passengers, and the water taxi was rated for a load limit of 3500 lb. Complete parts (a) through (d) below. a. Given that the water taxi was rated for a load limit of 3500 lb, what is the maximum mean weight of the passengers if the water taxi is filled to the stated capacity of 25 passengers? The maximum mean weight is 140 lb. (Type an integer or a decimal. Do not round.) b. If the water taxi is filled with 25 randomly selected passengers, what is the probability that their mean weight exceeds the value from part (a)? The probability is 1.0000. (Round to four decimal places as needed.) c. If the weight assumptions were revised so that the new capacity became 20 passengers and the water taxi is filled with 20 randomly selected passengers, 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? The probability is $\square$. (Round to four decimal places as needed.)
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Solution

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Solution Steps

Step 1: Maximum Mean Weight Calculation

To find the maximum mean weight (\(\mu_{max}\)) that the vehicle can accommodate, we use the formula \(\mu_{max} = \frac{L}{n}\), where \(L\) is the vehicle's load limit and \(n\) is its stated capacity. Substituting the given values, we get \(\mu_{max} = \frac{3500}{25} = 140\).

Step 2: Probability Calculation

To calculate the probability that the mean weight of a randomly selected group of passengers exceeds a specified value (\(\mu'\)), we first calculate the standard error of the mean (SE) using the formula \(SE = \frac{\sigma}{\sqrt{n'}}\), where \(\sigma\) is the standard deviation of passenger weights, and \(n'\) is the new capacity or number of passengers. Substituting the given values, we get \(SE = \frac{45}{\sqrt{20}} = 10.062\). Next, we calculate the Z-score for the specified mean weight (\(\mu'\)) using the formula \(Z = \frac{\mu' - \mu}{SE}\), where \(\mu\) is the original mean weight of passengers. Substituting the given values, we get \(Z = \frac{175 - 182}{10.062} = -0.696\). Finally, we use the Z-score to find the probability from the standard normal distribution table. The probability that the mean weight exceeds \(\mu'\) is 0.757.

Final Answer:

The maximum mean weight the vehicle can accommodate without exceeding its load limit is 140 kg. The probability that the mean weight of randomly selected passengers exceeds 175 kg is 0.757.

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