Question

last year over 12,000 students took an entrance exam at a certain state university. karens score was at the 69th percentile. rafaels score was at the 54th percentile.
(a) which of the following must be true about karens and rafaels scores?
rafael scored higher than karen.
both karen and rafael scored higher than the median.
karen answered 11 more questions correctly than rafael.
karens and rafaels scores were both in the bottom half of all scores.
(b) which of the following must be true about karens score?
karen scored lower than about 69% of the students who took the exam.
karen got about 69% of the questions correct.
karen scored higher than about 69% of the students who took the exam.
karen got about 69% of the questions wrong.

Explanation:

Part (a)

Step1: Recall the definition of percentile

A \( p^{th} \) percentile means that \( p\% \) of the data values are less than or equal to that value. The median is the \( 50^{th} \) percentile.
Karen is at the \( 69^{th} \) percentile and Rafael is at the \( 54^{th} \) percentile.
Since \( 69>54 \), Karen scored higher than Rafael.
The median is \( 50^{th} \) percentile. Since \( 69 > 50 \) and \( 54>50 \), both Karen and Rafael scored higher than the median.
We cannot say Karen answered 11 more questions correctly than Rafael because percentile is a relative - standing measure, not an absolute - count measure.
Since \( 69>50 \) and \( 54 > 50 \), their scores are not in the bottom half (bottom half is \( 0 - 49^{th} \) percentile)

Part (b)

Step1: Recall the definition of percentile

If Karen's score is at the \( 69^{th} \) percentile, by the definition of percentile, Karen scored higher than about \( 69\%\) of the students who took the exam.
Percentile is not about the percentage of questions correct or wrong. It is about the relative position of a score among all the scores.

Answer:

(a) Both Karen and Rafael scored higher than the median.
(b) Karen scored higher than about \( 69\%\) of the students who took the exam.