Question

last year over 8000 students took an entrance exam at a certain state university. mais score was at the 94th percentile. manuels score was at the 82nd percentile.
(if necessary, consult a list of formulas.)
(a) which of the following must be true about mais and manuels scores?
manuel scored higher than mai.
both mai and manuel scored higher than the median.
mai answered 12 more questions correctly than manuel.
mais and manuels scores were both in the bottom half of all scores.
(b) which of the following must be true about mais score?
mai scored lower than about 94% of the students who took the exam.
mai got about 94% of the questions correct.
mai scored higher than about 94% of the students who took the exam.
mai got about 94% of the questions wrong.

Explanation:

Part (a)

Step1: Recall the definition of percentile

The median is the 50th percentile. If a score is at the \(n\)th percentile, it means that approximately \(n\%\) of the scores are less than or equal to that score. Mai is at the 94th percentile and Manuel is at the 82nd percentile. Both 94 and 82 are greater than 50.

Step2: Analyze each option

• Option 1: Since \(94>82\), Mai scored higher than Manuel. So, “Manuel scored higher than Mai” is false.
• Option 2: The median is the 50th percentile. Since \(94 > 50\) and \(82>50\), both Mai and Manuel scored higher than the median.
• Option 3: Percentile is not directly related to the number of correct questions (it depends on the distribution of scores of all students). So, “Mai answered 12 more questions correctly than Manuel” is false.
• Option 4: Since \(94>50\) and \(82 > 50\), “Mai’s and Manuel’s scores were both in the bottom - half of all scores” is false.

Part (b)

Step1: Recall the definition of percentile

The \(n\)th percentile of a set of data is a value such that \(n\%\) of the data values are less than or equal to it.

Step2: Analyze each option

• Option 1: If Mai is at the 94th percentile, she scored higher than about \(94\%\) of the students, not lower. So, “Mai scored lower than about \(94\%\) of the students who took the exam” is false.
• Option 2: Percentile is about the relative position of the score among all scores, not the proportion of correct questions. So, “Mai got about \(94\%\) of the questions correct” is false.
• Option 3: By the definition of percentile, if Mai is at the 94th percentile, Mai scored higher than about \(94\%\) of the students who took the exam.
• Option 4: Percentile is not about the proportion of wrong questions. So, “Mai got about \(94\%\) of the questions wrong” is false.

Answer:

(a) Both Mai and Manuel scored higher than the median.
(b) Mai scored higher than about \(94\%\) of the students who took the exam.