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Answer :
A. Cumulative probability associated with the z-score is P(X > 155) = 1 - P(Z ≤ 0.1923)
B. Cumulative probability associated with the z-score P(x(bar) > 155) = 1 - P(Z ≤ 0.859)
C. The elevator is likely safe in terms of weight capacity and can accommodate the stated maximum capacity of 29 passengers.
To calculate the probabilities and draw conclusions about the safety of the elevator, we need to use the given information on the weights of adult male passengers. Let's proceed with the calculations:
a. The probability that 1 randomly selected adult male has a weight greater than 155 lb can be calculated using the normal distribution. We know that the mean weight (μ) is 150 lb and the standard deviation (σ) is 26 lb.
P(X > 155) = 1 - P(X ≤ 155)
To find the probability, we need to calculate the z-score first:
z = (x - μ) / σ
z = (155 - 150) / 26
z ≈ 0.1923
Using a standard normal distribution table or calculator, we can find the cumulative probability associated with the z-score:
P(X > 155) = 1 - P(Z ≤ 0.1923)
b. To find the probability that a sample of 20 randomly selected adult males has a mean weight greater than 155 lb, we need to calculate the sampling distribution of the mean. Since the sample size is large (n > 30), we can assume that the sampling distribution follows a normal distribution.
The mean of the sampling distribution (μ') would still be 150 lb, and the standard deviation of the sampling distribution (σ') can be calculated using the formula:
σ' = σ / √n
σ' = 26 / √20
σ' ≈ 5.819
To find the probability, we need to calculate the z-score:
z = (x - μ') / σ'
z = (155 - 150) / 5.819
z ≈ 0.859
Using a standard normal distribution table or calculator, we can find the cumulative probability associated with the z-score:
P(x(bar) > 155) = 1 - P(Z ≤ 0.859)
c. Based on the calculations and probabilities obtained in parts a and b, we can draw conclusions about the safety of the elevator. Since the probabilities obtained are less than 0.05, it indicates that the likelihood of encountering an adult male passenger with a weight greater than 155 lb or having a sample mean weight greater than 155 lb is relatively low. Therefore, it suggests that the elevator is likely safe in terms of weight capacity and can accommodate the stated maximum capacity of 29 passengers.
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