Examkin

Soru

Zorluk: Çok zorData Distributions and Measures

A quality control analyst records the battery life, in hours, of 11 distinct prototype laptop batteries. The shortest battery life recorded is 10 hours, the median battery life is 50 hours, and the range of the battery lives is 74 hours. If each battery life is a whole number of hours, what is the maximum possible mean battery life, in hours, of the 11 prototypes?

  1. A
    50
  2. 60Cevap
  3. C
    61
  4. D
    65

Cevap

The maximum possible mean battery life of the 11 prototypes is 60 hours.
The maximum possible mean is achieved by maximizing the sum of the 11 distinct integers under the given constraints. By setting the elements below the median to the largest possible distinct integers less than 50 (46, 47, 48, 49) and the elements above the median to the largest possible distinct integers less than the maximum value of 84 (80, 81, 82, 83), we find the maximum sum to be 660. Dividing this sum by the 11 elements yields a maximum mean of 60.

Adım Adım Çözüm

1
Define the variable terms representing the sorted battery lives.
Let the 11 battery lives in sorted order be x1<x2<x3<x4<x5<x6<x7<x8<x9<x10<x11x_1 < x_2 < x_3 < x_4 < x_5 < x_6 < x_7 < x_8 < x_9 < x_{10} < x_{11}.
Establishing the order of elements helps apply the median and distinct integer constraints.
2
Identify the values of the minimum, maximum, and median elements.
The minimum value is x1=10x_1 = 10. The median of 11 elements is the 6th element, so x6=50x_6 = 50. The range is 74, which means the maximum value is x11=10+74=84x_{11} = 10 + 74 = 84.
These fixed values act as boundaries for the remaining elements.
3
Maximize the remaining elements under the distinct integer constraint.
To maximize the mean, we must maximize the sum S=x1+x2+x3+x4+x5+x6+x7+x8+x9+x10+x11S = x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 + x_{10} + x_{11}. The elements below the median (x2,x3,x4,x5x_2, x_3, x_4, x_5) must be distinct integers strictly less than 50, so their maximum values are 46, 47, 48, and 49. The elements above the median (x7,x8,x9,x10x_7, x_8, x_9, x_{10}) must be distinct integers strictly less than 84, so their maximum values are 80, 81, 82, and 83.
Maximizing the individual elements maximizes the total sum, which in turn maximizes the mean.
4
Calculate the maximum sum and the resulting maximum mean.
The maximum sum is 10+46+47+48+49+50+80+81+82+83+84=66010 + 46 + 47 + 48 + 49 + 50 + 80 + 81 + 82 + 83 + 84 = 660. The maximum possible mean is 66011=60\frac{660}{11} = 60.
Dividing the maximum possible sum by the number of elements gives the maximum possible mean.

Anahtar Kavram

Maximizing the mean of a dataset with constraints on distinct values, median, and range.
Bu soruyu puanla

Konular

Data Distributions and Measures

Bu konudan daha fazla soru