Questions: Question 3, 6.4.12-T Part 1 of 3 A boat capsized and sank in a lake. Based on an assumption of a mean weight of 134 lb, the boat was rated to carry 50 passengers (so the load limit was 6,700 lb). After the boat sank, the assumed mean weight for similar boats was changed from 134 lb to 171 lb. Complete parts a and b below. a. Assume that a similar boat is loaded with 50 passengers, and assume that the weights of people are normally distributed with a mean of 174.3 lb and a standard deviation of 36.3 lb. Find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 134 lb. The probability is

Question 3, 6.4.12-T
Part 1 of 3
A boat capsized and sank in a lake. Based on an assumption of a mean weight of 134 lb, the boat was rated to carry 50 passengers (so the load limit was 6,700 lb). After the boat sank, the assumed mean weight for similar boats was changed from 134 lb to 171 lb. Complete parts a and b below.
a. Assume that a similar boat is loaded with 50 passengers, and assume that the weights of people are normally distributed with a mean of 174.3 lb and a standard deviation of 36.3 lb. Find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 134 lb.
The probability is

Transcript text: Question 3, 6.4.12-T Part 1 of 3 A boat capsized and sank in a lake. Based on an assumption of a mean weight of 134 lb, the boat was rated to carry 50 passengers (so the load limit was $6,700 \mathrm{lb}$). After the boat sank, the assumed mean weight for similar boats was changed from 134 lb to 171 lb. Complete parts a and b below. a. Assume that a similar boat is loaded with 50 passengers, and assume that the weights of people are normally distributed with a mean of 174.3 lb and a standard deviation of 36.3 lb. Find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 134 lb. The probability is $\square$

Solution

Solution Steps

Step 1: Given Information

We are given the following parameters for the weights of passengers:

Mean weight ($ \mu $) = 174.3 lb
Standard deviation ($ \sigma $) = 36.3 lb
Sample size ($ n $) = 50
Load limit mean weight = 134 lb

Step 2: Calculate Z-Score

To find the probability that the mean weight of 50 passengers exceeds 134 lb, we first calculate the Z-score for the lower bound (134 lb). The Z-score is calculated as follows:

\[ Z = \frac{X - \mu}{\sigma / \sqrt{n}} \]

Substituting the values:

\[ Z_{start} = \frac{134 - 174.3}{36.3 / \sqrt{50}} \approx -7.8502 \]

Step 3: Calculate Probability

Next, we calculate the probability that the mean weight is greater than 134 lb. This is given by:

\[ P(X > 134) = 1 - P(X \leq 134) = 1 - \Phi(Z_{start}) \]

Using the Z-score calculated:

\[ P(X > 134) = \Phi(\infty) - \Phi(-7.8502) = 1.0 - 0 \approx 1.0 \]