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Answer :
Final answer:
The score that separates the highest 15% from the rest in a normal distribution with a mean of 80 and standard deviation of 20 is 100.8, option C.
Explanation:
The question is asking to find the score that separates the top 15% from the rest in a normal distribution. To solve this, we look up the z-score that corresponds to the 85th percentile (100% - 15% = 85%) of a normal distribution, which is approximately 1.04. We then use the formula X = μ + (z)(σ), where X is the desired score, μ is the mean, and σ is the standard deviation. Plugging in the given mean (μ = 80) and standard deviation (σ = 20), we get X = 80 + (1.04)(20) which equals 100.8.
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Final answer:
The score that separates the highest 15% is approximately 72.3.
None of the given options is correct
Explanation:
To find the score that separates the highest 15% of the distribution from the rest, we can use the concept of z-scores and the standard normal distribution.
First, let's find the z-score corresponding to the highest 15% of the distribution. Since the normal distribution is symmetric, we know that 15% of the distribution lies above the mean (50% in total), leaving 35% below the mean.
Using a z-table or a calculator, we find that the z-score corresponding to 35% below the mean is approximately -0.385.
Next, we can use the z-score formula to find the corresponding score in the original distribution. The z-score formula is:
z = (X - μ) / σ
Rearranging the formula, we have:
X = μ + z * σ
Plugging in the values, we get:
X = 80 + (-0.385) * 20
X ≈ 80 - 7.7
X ≈ 72.3
Therefore, the score that separates the highest 15% of the distribution from the rest is approximately 72.3.
None of the answer choices provided match the correct score of approximately 72.3.
