Question

last year over 10,000 students took an entrance exam at a certain state university. melissa’s score was at the 59th percentile. boris’s score was at the 14th percentile. (if necessary, consult a list of formulas.) (a) which of the following must be true about melissa’s score? ○ melissa scored higher than about 41% of the students who took the exam. ○ about 41% of the students who took the exam scored higher than melissa. ○ melissa got about 59% of the questions correct. ○ melissa got about 59% of the questions wrong. (b) which of the following must be true about melissa’s and boris’s scores? ○ melissa scored higher than boris. ○ both melissa and boris scored higher than the median. ○ melissa got 45 more questions correct than boris. ○ melissa’s and boris’s scores were both in the bottom half of all scores.

Explanation:

Part (a)

Step 1: Recall Percentile Definition

A percentile (e.g., \( n \)-th percentile) means that \( n\% \) of the data is less than or equal to the value, and \( (100 - n)\% \) of the data is greater than the value. For Melissa's 59th percentile:

• \( 59\% \) of students scored less than or equal to her.
• \( 100 - 59 = 41\% \) of students scored higher than her.

Step 2: Evaluate Each Option

• Option 1: "Melissa scored higher than about 41%..." → False (She scored higher than \( 59\% \), lower than \( 41\% \)).
• Option 2: "About 41%...scored higher..." → True (Matches \( 100 - 59 = 41\% \)).
• Option 3: "Melissa got about 59% correct..." → False (Percentile ≠ percentage of correct answers).
• Option 4: "Melissa got about 59% wrong..." → False (Percentile ≠ percentage of wrong answers).

Step 1: Analyze Percentiles

• Melissa: 59th percentile (59% scored ≤ her, 41% > her).
• Boris: 14th percentile (14% scored ≤ him, 86% > him).

Step 2: Evaluate Each Option

• Option 1: "Melissa scored higher than Boris." → True (59th percentile > 14th percentile, so her score is higher).
• Option 2: "Both...higher than median." → False (Median is 50th percentile; Boris (14th) is below median).
• Option 3: "Melissa got 45 more questions..." → False (Percentile ≠ number of questions, no info on total questions).
• Option 4: "Both in bottom half..." → False (Melissa (59th) is in top half, Boris (14th) in bottom half).

Answer:

(a): About 41% of the students who took the exam scored higher than Melissa.

Part (b)