Scores on a recent national statistics exam were normally distributed with a mean of 80 and a standard deviation of 6.
What is the percentile rank of a score of 89?
Z = (score-mean)/SD
Look up Z score on table in back ofr stats text labeled something like "areas under the normal distribution" to get proportion corresponding to the percentile rank (proportion equal to or below that score).
12.5
To find the percentile rank of a score of 89, we need to determine the percentage of scores that fall below 89 in a normal distribution with a mean of 80 and a standard deviation of 6.
To do this, we can use a standard normal distribution table (also known as a z-table) or a statistical calculator. The z-score is a measure of how many standard deviations a particular score is above or below the mean. We can calculate the z-score using the formula:
z = (x - μ) / σ
Where:
- x is the given score (89)
- μ is the mean (80)
- σ is the standard deviation (6)
Substituting the values into the formula, we get:
z = (89 - 80) / 6
z = 9 / 6
z ≈ 1.5
Next, we need to find the area under the curve to the left of the z-score, which represents the percentage of scores below 89. This can be found using the z-table or a statistical calculator.
Using the z-table, we can look up the area corresponding to a z-score of 1.5. The z-table typically provides the area to the left of a given z-score, so we will find the closest z-score and corresponding area.
The closest z-score in the table is 1.50. The area to the left of 1.50 is 0.9332.
This means that approximately 93.32% of the scores fall below 89.
Therefore, the percentile rank of a score of 89 is approximately 93.32%.