Answer :
Final answer:
Using the Central Limit Theorem, the standard error is computed. Then, a Z-score is calculated with the formula Z = (X - μ) / (σ/√n). The Z-table or equivalent software yields a probability of about 0.046 or 4.6%.
Explanation:
To compute this, we need to apply the Central Limit Theorem which states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement,. Here, μ = 305 and σ = 46.5, and we have become large enough (n=125).
The formula for the standard deviation of the sampling distribution (standard error) is σ/√n. Here, it's 46.5/√125 = 4.155. We'll calculate a Z-score, which represents how many standard deviations an element is from the mean. Using the formula Z = (X - μ) / (σ/√n), where X is the score under question (298), we get -7/4.155 = -1.685.
Using a Z-table, or equivalently software with a function for the cumulative distribution of a normal distribution, we can find that the probability of observing a value more extreme is (1 - probability(Z < -1.685)) = 0.046. Therefore, the probability of selecting 125 observations and finding their mean is more than 298 is 0.046 or 4.6%.
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