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------------------------------------------------ A normal distribution has a mean of [tex]\mu = 80[/tex] and a standard deviation of [tex]\sigma = 20[/tex]. What score separates the highest [tex]15\%[/tex] of the distribution from the rest of the scores?

(Hint: two steps; think z-scores)

A. [tex]X = 95[/tex]
B. [tex]X = 59.2[/tex]
C. [tex]X = 100.8[/tex]
D. [tex]X = 65[/tex]

To find the score that separates the highest 15% of the distribution from the rest, follow these steps:

1. Understand the Problem: We want to find the score in a normal distribution with a mean (μ) of 80 and a standard deviation (σ) of 20 that separates the highest 15% from the rest. This corresponds to the 85th percentile, as 100% - 15% = 85%.

2. Find the z-score: Use a z-table or a statistical tool to find the z-score that corresponds to the 85th percentile of a standard normal distribution. This z-score is approximately 1.04.

3. Use the z-score Formula: The formula to convert a z-score to a raw score (X) is:
[tex]\[ X = \mu + (z \times \sigma) \][/tex]
Plug in the values:
- μ = 80 (mean)
- z ≈ 1.04 (z-score for the 85th percentile)
- σ = 20 (standard deviation)

4. Calculate the Raw Score (X):
[tex]\[ X = 80 + (1.04 \times 20) \][/tex]
[tex]\[ X = 80 + 20.8 \][/tex]
[tex]\[ X = 100.8 \][/tex]

5. Conclusion: The score that separates the highest 15% of the distribution from the rest is 100.8. From the given options, the correct choice is [tex]$ X = 100.8 $[/tex].