Answer :
The value that separates the bottom 15% of the population from the top 85% is approximately -66.7, while the sample mean that separates the bottom 15% from the top 85% is approximately 9.6.
The value that separates the bottom 15% of values from the top 85% in a normal distribution can be found using the z-score associated with the 15th percentile. Using a standard normal distribution table, the z-score associated with 15% is approximately -1.036. Therefore, you can use the formula X = μ + Zσ, where X is the value you're trying to find, μ is the mean, Z is the z-score, and σ is the standard deviation. After plugging in the supplied values, X = 35.9 - 1.036 * 98.2 = -66.7 (approximately).
Next, we need to find the sample mean separating the bottom 15% from the top 85%. For this, we will again use the z-score method, but adjust the standard deviation because the standard deviation of the sampling distribution of the mean is σ/√n (Central Limit Theorem). Thus, X = μ + z(σ/√n) = 35.9 - 1.036 * (98.2/√21) = 9.6 (approximately).
Learn more about Z-score Method here:
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