Math Problem Statement

The table below shows the number of patients randomized in a clinical trial in each of the first five months of the study. If the median enrollment for the first five months was 7, what is the smallest (i.e., least) possible value for n?

Solution

The problem asks for the smallest possible value of nn (the number of patients randomized in the first month) such that the median enrollment for the first five months is 7.

Step-by-step approach:

  1. The number of patients randomized in the first five months is n,9,7,2,3n, 9, 7, 2, 3.
  2. The median is the middle value when the data is ordered from least to greatest.
  3. Currently, the data set is n,9,7,2,3n, 9, 7, 2, 3. Sorting this (once nn is determined) will help us determine the median.

Since there are 5 values, the median is the 3rd value in the ordered list.

  1. The condition is that the median should be 7. So, when the data is arranged in ascending order, the third value must be 7.

Let's determine the smallest possible value of nn by testing different values and observing the order of the data.

Test different values of nn:

  • If n=6n = 6: The data set becomes 6,9,7,2,36, 9, 7, 2, 3.

    • Sorting: 2,3,6,7,92, 3, 6, 7, 9. The median is 6, not 7.
  • If n=7n = 7: The data set becomes 7,9,7,2,37, 9, 7, 2, 3.

    • Sorting: 2,3,7,7,92, 3, 7, 7, 9. The median is 7, which satisfies the condition.

Thus, the smallest possible value of nn is 6\boxed{6}.

Would you like further explanation or have any questions?


Here are 5 related questions for further practice:

  1. What happens if the median is required to be 8 instead of 7?
  2. How would the value of nn change if there were 6 months of data?
  3. What is the mean of the set n,9,7,2,3n, 9, 7, 2, 3 for n=7n = 7?
  4. How do you calculate the range of a data set?
  5. What is the mode of the data set n,9,7,2,3n, 9, 7, 2, 3 for n=7n = 7?

Tip: When dealing with medians, always sort the data set first to easily locate the middle value!

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Math Problem Analysis

Mathematical Concepts

Median Calculation
Statistics
Data Sorting

Formulas

Median of Odd Set of Numbers: Middle value in ordered data

Theorems

Median Theorem in Statistics

Suitable Grade Level

Grades 9-11