Final answer:
To find the 85th percentile of a normal population with a mean of 51 and a standard deviation of 7, use the z-score for 0.85 and apply the formula: percentile value = (z-score x standard deviation) + mean. The z-score approximates to 1.04, thus the 85th percentile is approximately 58.28, closest to answer option a) 58.13.
Step-by-step explanation:
The question asks about finding the 85th percentile of a normal population with a mean of 51 and a standard deviation of 7. To solve this, we need to use the standard normal distribution (z-distribution) since the percentile indicates a value below which a certain percent of observations fall.
To find the 85th percentile, we look for the z-score that corresponds to 0.85 in the z-table (or use a calculator that includes the normal distribution functions). This z-score tells us how many standard deviations from the mean our value is. The formula to then find the percentile value is:
Percentile Value = (z-score × standard deviation) + mean
In this case, the z-score that corresponds to the 85th percentile is approximately 1.04. So, we would compute the following:
Percentile Value = (1.04 × 7) + 51
Percentile Value = 7.28 + 51
Percentile Value = 58.28
This value rounds off to 58.28, which would be closest to 58.13. Therefore, the correct answer is option a) 58.13.