A ferry will safely accommodate 68 tons of passenger cars. Assume that the mean weight of a passenger car is 1.8 tons with standard deviation 0.5 tons. If a random sample of 35 cars are loaded onto the ferry, what is the probability that the maximum safe weight will be exceeded?
The Correct answer and Explanation is :
The correct answer is: P(X>68)=1−0.9545≈0.0455
To find the probability that the total weight of 35 randomly selected passenger cars exceeds the ferry’s capacity of 68 tons, we can use the Central Limit Theorem (CLT). The CLT states that, for large enough sample sizes, the distribution of the sample mean will approach a normal distribution, regardless of the shape of the population distribution.
Step 1: Define Parameters
- Mean weight of a passenger car (( \mu )): 1.8 tons
- Standard deviation of the weight (( \sigma )): 0.5 tons
- Sample size (( n )): 35 cars
Step 2: Calculate the Sampling Distribution
- Mean of the Sampling Distribution:
[
\mu_{\bar{x}} = \mu \times n = 1.8 \text{ tons}
] - Standard Deviation of the Sampling Distribution (Standard Error):
[
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{0.5}{\sqrt{35}} \approx 0.0844 \text{ tons}
]
Step 3: Calculate the Total Weight
To find the probability that the total weight exceeds 68 tons, we first convert this into the equivalent mean for the sample of 35 cars:
[
\text{Total weight limit} = 68 \text{ tons}
]
[
\text{Mean weight limit per car} = \frac{68}{35} \approx 1.943 \text{ tons}
]
Step 4: Standardize
Next, we standardize this value to find the z-score:
[
z = \frac{\bar{x} – \mu_{\bar{x}}}{\sigma_{\bar{x}}} = \frac{1.943 – 1.8}{0.0844} \approx 1.693
]
Step 5: Find the Probability
Using a z-table or a calculator, we can find the probability corresponding to ( z = 1.693 ). This gives us the area to the left of the z-score.
- The area to the left of ( z = 1.693 ) is approximately ( 0.9545 ). Therefore, the probability that the total weight exceeds 68 tons is:
[
P(X > 68) = 1 – 0.9545 \approx 0.0455
]
Conclusion
Thus, the probability that the maximum safe weight of the ferry will be exceeded when loading 35 cars is approximately ( 0.0455 ) or ( 4.55\% ). This indicates a relatively low likelihood of exceeding the weight limit, meaning the ferry can safely accommodate the given sample of passenger cars with a high probability.