To determine the probability that the combined weight of 60 adult men riding the roller coaster will exceed the maximum safe weight of 12,000 pounds, we can use the central limit theorem (CLT).
Step 1: Calculate the mean and standard deviation of the sample weights.
The mean weight of adult U.S. men is given as \( \mu = 193 \) pounds, and the standard deviation as \( \sigma = 67 \) pounds.
Step 2: Calculate the mean and standard deviation for the total weight of 60 men.
When considering a sample of \( n = 60 \):
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The mean total weight (\( \mu_{total} \)): \[ \mu_{total} = n \times \mu = 60 \times 193 = 11580 \text{ pounds} \]
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The standard deviation of the total weight (\( \sigma_{total} \)): \[ \sigma_{total} = \sqrt{n} \times \sigma = \sqrt{60} \times 67 \approx 8.164 \times 67 \approx 547.00 \text{ pounds} \]
Step 3: Find the z-score for the maximum safe weight.
To find the probability that the combined weight exceeds 12,000 pounds, we first calculate the z-score:
\[ z = \frac{X - \mu_{total}}{\sigma_{total}} = \frac{12000 - 11580}{547.00} = \frac{420}{547.00} \approx 0.7677 \]
Step 4: Look up the z-score in the standard normal distribution table.
Now, we look up \( z = 0.7677 \) in the standard normal distribution table, or use a calculator to find the cumulative probability:
Using a standard normal table or calculator, we find: \[ P(Z < 0.7677) \approx 0.7764 \]
Step 5: Find the probability that the total weight exceeds 12,000 pounds.
To find the probability that the weight exceeds 12,000 pounds, we need to subtract the cumulative probability from 1:
\[ P(X > 12000) = 1 - P(Z < 0.7677) \approx 1 - 0.7764 = 0.2236 \]
Final Answer: The probability that the maximum safe weight will be exceeded if 60 adult men ride the coaster is approximately \( \boxed{0.2236} \).
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