Answer :
To find the probability that the mean weight of 12 adult male passengers is greater than 169 pounds, we can use the central limit theorem. The probability of the mean weight exceeding 169 pounds is approximately 0.1333 or 13.33%.
The central limit theorem states that the distribution of sample means approaches a normal distribution as the sample size increases. Since the weights of the adult male passengers are normally distributed with a mean of 173 pounds and a standard deviation of 29 pounds, we can calculate the z-score for a mean weight of 169 pounds. The z-score formula is given by z = (X - μ) / (σ / sqrt(n)), where X is the mean weight, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Plugging in the values, we have z = (169 - 173) / (29 / sqrt(12)).
Calculating this, we get z = -1.103. Using a standard normal distribution table or calculator, we can find the probability associated with this z-score.
The probability of having a mean weight greater than 169 pounds is the same as the probability of having a z-score less than -1.103, which is approximately 0.1333 or 13.33%.
Since the probability of the mean weight exceeding 169 pounds is less than 50%, it is unlikely that the elevator will be overloaded when loaded with 12 adult male passengers. Therefore, this elevator appears to be safe.
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