Examkin

Soru

Zorluk: Çok zorMean, Median, and Mode

A data set SS consists of 99 positive integers arranged in non-decreasing order: x1≤x2≤x3≤x4≤x5≤x6≤x7≤x8≤x9x_1 \le x_2 \le x_3 \le x_4 \le x_5 \le x_6 \le x_7 \le x_8 \le x_9. The arithmetic mean of set SS is 2222, the median is 2020, and the set has a unique mode of 1515. If the range of set SS is 3030, what is the maximum possible value of the largest element, x9x_9

  1. A
    31
  2. B
    40
  3. 45Cevap
  4. D
    50
  5. E
    52

Cevap

The maximum possible value of the largest element is 45.
To maximize the largest value in a set with a fixed range of 30, we must maximize the smallest value because the maximum value equals the minimum value plus 30. Since 15 is the unique mode of the ordered set, 15 must be an element of the set, which restricts the minimum value to at most 15. Setting the smallest element to 15 allows the largest element to reach 15 + 30 = 45, which can be verified to satisfy all mean, median, and mode constraints.

Adım Adım Çözüm

1
Relate total sum, range, and median to the elements of the set.
The sum of all 9 elements is 9×22=1989 \times 22 = 198. For a 9-element set ordered as x1≤x2≤⋯≤x9x_1 \le x_2 \le \dots \le x_9, the median is the 5th element x5=20x_5 = 20. The range is x9−x1=30x_9 - x_1 = 30, so x9=x1+30x_9 = x_1 + 30.
Establishing the basic relationships among the statistical measures given in the problem.
2
Determine the upper bound for the smallest element x1x_1.
Because 1515 is the unique mode of the set, 1515 must be present in the set. Since the set is in non-decreasing order, x1≤15x_1 \le 15.
If x1>15x_1 > 15, then no element in the set could equal 1515, violating the condition that 1515 is the mode.
3
Maximize x9x_9 using the relationship x9=x1+30x_9 = x_1 + 30.
To maximize x9x_9, x1x_1 must be as large as possible. The maximum possible value for x1x_1 is 1515, which gives x9=15+30=45x_9 = 15 + 30 = 45.
Connecting the range equation to the upper bound on x1x_1.
4
Verify that a valid set exists for x1=15x_1 = 15 and x9=45x_9 = 45.
Consider the set {15,15,15,15,20,20,25,28,45}\{15, 15, 15, 15, 20, 20, 25, 28, 45\}. Sum =198= 198, Mean =22= 22, Median =20= 20 (5th element), Mode =15= 15 (frequency 4), Range =45−15=30= 45 - 15 = 30. All conditions are satisfied.
Ensuring the upper bound is achievable under all given statistical constraints.

Anahtar Kavram

Interplay between statistical measures (mean, median, mode, range) and set boundaries
Bu soruyu puanla

Konular

Mean, Median, and Mode

Bu konudan daha fazla soru