A ferry will safely accommodate 66 tons of passenger cars. Assume that the mean weight of a passenger car is 1.7 tons with standard deviation 0.5 tons. If a random sample of 36 cars are loaded onto the ferry, what is the probability that the maximum safe weight will be exceeded?
The Correct Answer and Explanation is :
To determine the probability that the total weight of 36 randomly selected passenger cars exceeds the ferry’s capacity of 66 tons, we can model the total weight as the sum of 36 independent random variables, each representing the weight of a single car.
Step 1: Define the Random Variables
Let ( X_i ) represent the weight of the ( i )-th car. Each ( X_i ) is a random variable with:
- Mean (( \mu )) = 1.7 tons
- Standard deviation (( \sigma )) = 0.5 tons
The total weight ( W ) of the 36 cars is the sum of these individual weights:
[ W = X_1 + X_2 + \cdots + X_{36} ]
Step 2: Calculate the Mean and Standard Deviation of the Total Weight
The mean of the total weight is the sum of the means of the individual weights:
[ \mu_W = 36 \times 1.7 = 61.2 \, \text{tons} ]
The standard deviation of the total weight is the square root of the sum of the variances of the individual weights. Since the weights are independent, the variances add:
[ \sigma_W = \sqrt{36 \times (0.5)^2} = \sqrt{36 \times 0.25} = \sqrt{9} = 3 \, \text{tons} ]
Step 3: Standardize the Total Weight
To find the probability that the total weight exceeds 66 tons, we first standardize the value 66 tons to a z-score using the formula:
[ z = \frac{X – \mu_W}{\sigma_W} ]
Substituting the values:
[ z = \frac{66 – 61.2}{3} = \frac{4.8}{3} \approx 1.6 ]
Step 4: Find the Probability
The z-score of 1.6 corresponds to the area to the left of 66 tons under the normal distribution curve. Using standard normal distribution tables or a calculator, we find:
[ P(Z \leq 1.6) \approx 0.9452 ]
Therefore, the probability that the total weight exceeds 66 tons is:
[ P(Z > 1.6) = 1 – P(Z \leq 1.6) = 1 – 0.9452 = 0.0548 ]
Thus, there is approximately a 5.48% chance that the total weight of the 36 cars will exceed the ferry’s capacity of 66 tons.
Conclusion
By modeling the total weight of the cars as the sum of independent random variables and applying the properties of the normal distribution, we determined that the probability of exceeding the ferry’s capacity is about 5.48%. This analysis is crucial for ensuring the safety and operational efficiency of the ferry.