Unit 2 — Advanced Measurements in Statistics · Statistics. These are 10 questions from the Magmo3 bank on this lesson, from easy to hard. Solve each one, then open "Show answer and explanation". Magmo3 has 111 questions on this lesson, each with an explanation.
If the five values of a box plot, in order, are 15, 32, 40, 49, 58, then the upper quartile = ……
Correct answer: (D) 49
Explanation: Step 1: The five values are arranged as: minimum value, lower quartile, median, upper quartile, maximum value. Step 2: minimum = 15, lower quartile = 32, median = 40, upper quartile = 49, maximum = 58 ∴ upper quartile = 49
The first quartile (Q1) of a set of ordered data is the value that is preceded by …… of the data.
Correct answer: (C) 25%
Explanation: Step 1: Q1 is the value preceded by a quarter of the data (25%) and followed by three-quarters of the data. Step 2: 50% belongs to the second quartile Q2 (the median) and 75% to the third quartile Q3. ∴ 25%
In a box plot, a rectangle (the box) is drawn that begins at …… and ends at the upper quartile.
Correct answer: (A) the lower quartile
Explanation: Step 1: The box plot draws a rectangle that begins at the lower quartile and ends at the upper quartile. Step 2: Lines extend from the box to the minimum value and to the maximum value. ∴ the lower quartile
In a frequency distribution with classes 0–, 4–, 8–, 12–, 16– and frequencies 5, 8, 12, 10, 5 respectively, the third quartile lies in the class ……
Correct answer: (C) 12–
Explanation: Step 1: total frequency = 40, so order of Q3 = 3 × 40 ÷ 4 = 30 Step 2: ascending cumulative frequencies: 5, 13, 25, 35; the order 30 lies between 25 and 35 ∴ the class 12– (the class 8– contains the median, order 20).
In an exam taken by 200 students, the expected number of students whose marks lie between the lower quartile and the maximum value in the box plot of their marks = ……
Correct answer: (B) 150
Explanation: Step 1: The lower quartile is preceded by 25% of the data, so 75% of the data follow it. Step 2: 75% × 200 = 150 ∴ 150 students (100 is the number inside the box only, between the two quartiles).
The first quartile of the values 14, 20, 9, 25, 31, 17, 28, 12, 22, 35 is ……
Correct answer: (C) 13.5
Explanation: Step 1: In order: 9, 12, 14, 17, 20, 22, 25, 28, 31, 35 Step 2: order of Q1 = (10 + 1) ÷ 4 = 2.75, between the 2nd value (12) and the 3rd value (14) Step 3: Q1 = 12 + (14 − 12) × (2.75 − 2) = 13.5 ∴ Q1 = 13.5 (12.5 comes from multiplying by 0.25 instead of 0.75).
The leaves in the following stem-and-leaf plot are in ascending order (Key 34 = 3|4): stem 2: 3, 5, 8 — stem 3: 1, 4, x, 7 — stem 4: 0, 2, 6, 9 — stem 5: 3. If the mode of these data = 37, then Q3 − Q2 = ……
Correct answer: (A) 8
Explanation: Step 1: The mode 37 means 37 is repeated; the leaf x is between 4 and 7, so x = 7 Step 2: n = 12, values: 23, 25, 28, 31, 34, 37, 37, 40, 42, 46, 49, 53. Order of Q2 = 6.5, so Q2 = 37 Step 3: order of Q3 = 3 × 13 ÷ 4 = 9.75, so Q3 = 42 + 0.75 × 4 = 45 ∴ Q3 − Q2 = 45 − 37 = 8 (taking x = 4 makes Q2 = 35.5 and the difference 9.5).
The following table shows the percentages of the ages of the employees of a company: classes 10–, 20–, 30–, 40–, 50–60 with percentages 10%, 20%, 30%, 25%, 15% respectively. Without knowing the number of employees, Q3 − Q1 for their ages = ……
Correct answer: (C) 18.5
Explanation: Step 1: Treat the percentages as frequencies with total 100. Order of Q1 = 25, in the interval 20– with cumulative frequency 10 at its start: Q1 = 20 + (25 − 10) ÷ 20 × 10 = 27.5 Step 2: order of Q3 = 75, in the interval 40– with cumulative frequency 60 at its start: Q3 = 40 + (75 − 60) ÷ 25 × 10 = 46 Step 3: Q3 − Q1 = 46 − 27.5 = 18.5 ∴ 18.5 (it does not depend on the number of employees, because the orders and the frequencies change in the same ratio).
The ages, in years, of the 9 members of a family are 12, 3, 45, 7, 70, 15, 41, 10, 38. After 4 years a new baby is born into the family. At that time, the first quartile of the ages of the ten family members = ……
Correct answer: (C) 10
Explanation: Step 1: After 4 years every age increases by 4 and the newborn's age 0 is added: in order 0, 7, 11, 14, 16, 19, 42, 45, 49, 74 Step 2: n = 10, so order of Q1 = 11 ÷ 4 = 2.75: Q1 = 7 + 0.75 × (11 − 7) = 10 ∴ Q1 = 10 (8.5 is Q1 of the original nine ages, and 12.5 is that value increased by 4 when the newborn is forgotten).
A double stem-and-leaf plot shows the marks of the students of two classes (Key 50 = 5|0): stem 3: first class 5, 8 and second class 1, 4, 9 — stem 4: first class 1, 2, 6, 7 and second class 0, 5, 5 — stem 5: first class 0, 3, 9 and second class 2, 6, 8 — stem 6: first class 2 and second class 0, 3. Upper quartile of the first class − lower quartile of the second class = ……
Correct answer: (A) 15.5
Explanation: Step 1: First class (10 values): order of Q3 = 3(10 + 1) ÷ 4 = 8.25, between 53 and 59: Q3 = 53 + 6 × 0.25 = 54.5 Step 2: Second class (11 values): n + 1 = 12 is divisible by 4, so order of Q1 = 3 and Q1 = 39 Step 3: 54.5 − 39 = 15.5 ∴ 15.5 (14 comes from taking Q3 of the first class as 53 directly, and 18.5 from multiplying by 0.75 instead of 0.25).
The first 10 questions in every subject are free, each with an explanation.