Examkin

Question

Difficulty: Very hardDescriptive Statistics and Data Representations

A dataset consists of 77 positive integers sorted in non-decreasing order. The mean of the dataset is 2020, the median is 1818, and the unique mode is 1515. If the range of the dataset is 2222, what is the minimum possible value of the largest number in the dataset?

  1. A
    27
  2. B
    29
  3. 30Answer
  4. D
    35
  5. E
    37

Answer

30
The correct answer is 30 because minimizing the largest value requires minimizing the smallest value under the sum and uniqueness constraints. When the smallest value is 8, the largest value is 30, which allows for a valid, sorted sequence of positive integers where 15 is the unique mode: {8, 15, 15, 18, 26, 28, 30}.

Step-by-Step Solution

1
Set up the variables for the sorted dataset.
Let the seven positive integers in non-decreasing order be x1,x2,x3,x4,x5,x6,x7x_1, x_2, x_3, x_4, x_5, x_6, x_7. Since the median is the 4th value, x4=18x_4 = 18. The mean is 2020, so the sum of all elements is 7×20=1407 \times 20 = 140.
Establishing standard notation and utilizing the definitions of median and mean.
2
Incorporate the range into the sum equation.
x1+x2+x3+18+x5+x6+x7=140x_1 + x_2 + x_3 + 18 + x_5 + x_6 + x_7 = 140. Since the range is 2222, the largest number is x7=x1+22x_7 = x_1 + 22. Substituting this into the sum gives 2x1+x2+x3+x5+x6=1002x_1 + x_2 + x_3 + x_5 + x_6 = 100.
Simplifying the system of equations by expressing the largest element in terms of the smallest element.
3
Apply the mode constraint.
The unique mode is 1515. Since x4=18x_4 = 18 and the dataset is sorted, 1515 must be in the lower half of the dataset and must repeat. Thus, x2=x3=15x_2 = x_3 = 15. The equation simplifies to 2x1+30+x5+x6=100  ⟹  2x1+x5+x6=702x_1 + 30 + x_5 + x_6 = 100 \implies 2x_1 + x_5 + x_6 = 70.
Using the properties of the mode and sorted list to lock the values of the second and third elements.
4
Set up inequalities to minimize the largest value.
To minimize x7=x1+22x_7 = x_1 + 22, we must minimize x1x_1, which is equivalent to maximizing x5x_5 and x6x_6. Since 1515 is the unique mode with a frequency of 22, no other value can repeat. Thus, x5,x6,x_5, x_6, and x7x_7 must be distinct from each other and larger than the median 1818. This means x5≥19x_5 \ge 19. The upper bounds are x6≤x7−1=x1+21x_6 \le x_7 - 1 = x_1 + 21 and x5≤x7−2=x1+20x_5 \le x_7 - 2 = x_1 + 20.
Applying the uniqueness constraint of the mode to restrict the values of the upper half of the dataset.
5
Solve for the minimum integer value of the smallest element.
Substitute the maximum bounds of x5x_5 and x6x_6 into the equation: 2x1+(x1+20)+(x1+21)≥70  ⟹  4x1+41≥70  ⟹  4x1≥29  ⟹  x1≥7.252x_1 + (x_1 + 20) + (x_1 + 21) \ge 70 \implies 4x_1 + 41 \ge 70 \implies 4x_1 \ge 29 \implies x_1 \ge 7.25. Since x1x_1 must be an integer, the minimum possible value is x1=8x_1 = 8. This yields a minimum largest value of x7=8+22=30x_7 = 8 + 22 = 30.
Solving the inequality system to find the smallest valid integer boundaries.

Key Concept

Descriptive Statistics and Optimization
Rate this question

Topics

Descriptive Statistics and Data Representations

More questions from this topic