Question

last year over 7000 students took an entrance exam at a certain state university. lucys score was at the 91st percentile. miguels score was at the 31st percentile.
(if necessary, consult a list of formulas.)
(a) which of the following must be true about lucys and miguels scores?
both lucy and miguel scored higher than the median.
both lucy and miguel scored below than the median.
lucy scored higher than miguel.
miguel scored higher than lucy.
(b) which of the following must be true about lucys score?
about 91% of the students who took the exam scored lower than lucy.
lucy got about 91% of the questions correct.
lucys score was in the bottom half of all scores.
lucy missed 9 questions.

Explanation:

Part (a)

Step 1: Recall Percentile and Median

The median is the 50th percentile (middle value of ordered data). Lucy is at 91st percentile (91% scores ≤ her score), Miguel at 31st (31% scores ≤ his score).

Step 2: Analyze Each Option

• "Both higher than median": Median is 50th. Lucy (91st) > 50th, Miguel (31st) < 50th. False.
• "Both below median": Lucy > 50th. False.
• "Lucy scored higher than Miguel": 91st percentile > 31st percentile (more scores ≤ Lucy’s, so her score is higher). True.
• "Miguel scored higher than Lucy": Contradicts percentile order. False.

Part (b)

Step 1: Define Percentile

The \( n \)-th percentile means \( n\% \) of data is less than or equal to that value. Lucy is at 91st percentile.

Step 2: Analyze Each Option

• "About 91% scored lower than Lucy": Matches percentile definition (91% of scores ≤ her score, so ~91% scored lower). True.
• "91% questions correct": Percentile ≠ percentage of correct answers. False.
• "Bottom half (≤50th percentile)": 91st > 50th. False.
• "Missed 9 questions": Percentile doesn’t relate to missed questions. False.

Final Answers:

(a) Lucy scored higher than Miguel.
(b) About 91% of the students who took the exam scored lower than Lucy.

Answer:

Step 1: Define Percentile

The \( n \)-th percentile means \( n\% \) of data is less than or equal to that value. Lucy is at 91st percentile.

Step 2: Analyze Each Option

• "About 91% scored lower than Lucy": Matches percentile definition (91% of scores ≤ her score, so ~91% scored lower). True.
• "91% questions correct": Percentile ≠ percentage of correct answers. False.
• "Bottom half (≤50th percentile)": 91st > 50th. False.
• "Missed 9 questions": Percentile doesn’t relate to missed questions. False.

Final Answers:

(a) Lucy scored higher than Miguel.
(b) About 91% of the students who took the exam scored lower than Lucy.