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Answer :
Final answer:
The probability that 1 randomly selected adult male has a weight greater than 152lb is approximately 0.8121, and the probability that a sample of 27 randomly selected adult males has a mean weight greater than 152lb is approximately 0.0092.
Explanation:
To find the probability that 1 randomly selected adult male has a weight greater than 152lb, we need to calculate the z-score and then find the area under the standard normal curve beyond that z-score.
The z-score is calculated as (x - μ) / σ, where x is the weight, μ is the mean, and σ is the standard deviation. Plugging in the values, we have (152 - 176) / 27 = -0.8889.
Using a standard normal table or a calculator, we can find that the area to the right of -0.8889 is approximately 0.8121. Therefore, the probability that 1 randomly selected adult male has a weight greater than 152lb is 0.8121.
To calculate the probability that a sample of 27 randomly selected adult males has a mean weight greater than 152lb, we use the Central Limit Theorem. The Central Limit Theorem states that for a large enough sample size, the distribution of the sample mean approaches a normal distribution.
The mean of the sample mean is still 176lb, but the standard deviation is σ / √n, where σ is the population standard deviation and n is the sample size. Plugging in the values, we have 27lb / √27 = 5.1962.
Again, we calculate the z-score using the formula (x - μ) / (σ / √n). Plugging in the values, we have
(152 - 176) / (27 / √27) = -2.3664.
Using a standard normal table or a calculator, we can find that the area to the right of -2.3664 is approximately 0.0092. Therefore, the probability that a sample of 27 randomly selected adult males has a mean weight greater than 152lb is 0.0092.
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