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Loading a Tour Boat The Ethan Allen tour boat capsized and sank in Lake George, New York, and 20 of the 47 passengers drowned. Based on a 1960 assumption of a mean weight of 140 lb for passengers, the boat was rated to carry 50 passengers. After the boat sank, New York State changed the assumed mean weight from 140 lb to 174 lb.

a. Given that the boat was rated for 50 passengers with an assumed mean of 140 lb, the boat had a passenger load limit of 7000 lb. Assume that the boat is loaded with 50 male passengers, and 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 鈥淏ody Data鈥� in Appendix B). Find the probability that the boat is overloaded because the 50 male passengers have a mean weight greater than 140 lb.

b. The boat was later rated to carry only 14 passengers, and the load limit was changed to 2436 lb. If 14 passengers are all males, find the probability that the boat is overloaded because their mean weight is greater than 174 lb (so that their total weight is greater than the maximum capacity of 2436 lb). Do the new ratings appear to be safe when the boat is loaded with 14 male passengers?

Step by step solution

01

Given information

The weights of the men are normally distributed with a mean $\left(渭\right)$equal to 189lb and a standard deviation $\left(蟽\right)$equal to 39 lb.

02

Required probabilities

a.

Let $\stackrel{炉}{x}$denote the sample mean weight of the men.

The sample mean weight of the men follows a normal distribution with a mean equal to ${渭}_{\stackrel{炉}{x}}=渭$and a standard deviation equal to ${蟽}_{\stackrel{炉}{x}}=\frac{蟽}{\sqrt{n}}$${蟽}_{\stackrel{炉}{x}}=\frac{蟽}{\sqrt{n}}$.

The sample size is equal to n=50.

The probability that the sample mean weight of the men is greater than 140 lb is computed using the standard normal table, as shown below.

$$\begin{gathered} P\left(\stackrel{炉}{x}>140\right)=P\left(\stackrel{炉}{x}>140\right) \\ =P\left(\frac{\stackrel{炉}{x}-渭}{\frac{蟽}{\sqrt{n}}}>\frac{140-渭}{\frac{蟽}{\sqrt{n}}}\right) \\ =P\left(z>\frac{140-189}{\frac{39}{\sqrt{50}}}\right) \end{gathered}$$

$$\begin{gathered} =P\left(z>-8.88\right) \\ =P\left(z<8.88\right) \\ =1.0000 \end{gathered}$$

Therefore, the probability that the boat is overloaded or that the sample mean weight of the men is greater than 140 lb is equal to 1.0000.

b.

The sample size is now equal to n=14.

The probability that the sample mean weight of the men is greater than 174 is computed using the standard normal table, as shown below.

$$\begin{gathered} P\left(\stackrel{炉}{x}>174\right)=P\left(\stackrel{炉}{x}>174\right) \\ =P\left(\frac{\stackrel{炉}{x}-渭}{\frac{蟽}{\sqrt{n}}}>\frac{174-渭}{\frac{蟽}{\sqrt{n}}}\right) \\ =P\left(z>\frac{174-189}{\frac{39}{\sqrt{14}}}\right) \end{gathered}$$

$$\begin{gathered} =P\left(z>-1.44\right) \\ =P\left(z<1.44\right) \\ =0.9251 \end{gathered}$$

Therefore, the probability that the boat is overloaded or that the sample mean weight of the men is greater than 174 lb is equal to 0.9251.

Yes, the new ratings appear to be safe as the probability of the boat being overloaded is smaller as compared to the previous ratings.

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