A scientist surveyed 15 different forest regions and recorded the number of endangered pla… — SAT

A scientist surveyed $15$ different forest regions and recorded the number of endangered plant species in each region. The numbers of endangered species in $14$ of the regions are:

12, 15, 15, 16, 18, 20, 20, 22, 22, 22, 25, 28, 30, 34

The number of endangered species in the $15\text{th}$ region is $x$ , where $x$ is an integer greater than $25$ . If the median of the numbers of endangered species for all $15$ regions is equal to the mean, what is the value of $x$ ?

Answer: 31

Answer

The correct answer is 31.

The correct answer is 31. The sum of the 14 known values is 299. Since the 15th value

x

is greater than 25, it must be placed at or after the 12th position when all 15 values are sorted. Thus, the 8th value (the median of the 15 values) remains the same as the 8th value of the known 14 sorted values, which is 22. Setting the mean equal to the median gives

\frac{299 + x}{15} = 22

, which simplifies to

299 + x = 330

, resulting in

x = 31

. Since

31 > 25

, this is the correct value.

Step-by-Step Solution

Find the sum of the 14 known values.

The sum of the 14 known values is 299.

This is needed to construct the expression for the mean of the 15 values.

Determine the median of the 15 values given the constraint

x > 25

The median is 22.

Since

x > 25

, it will be sorted after the first 11 values (the 11th value is 25). The median of 15 values is the 8th value, which remains 22.

Set the mean equal to the median and solve for

x

\frac{299 + x}{15} = 22 \implies 299 + x = 330 \implies x = 31

This yields the value of

x

that satisfies the condition that the mean equals the median.

Key Concept

Calculating and comparing the mean and median of a dataset, and analyzing how a new data point affects these measures.

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