A biology student recorded the mass, in grams, of 6 plant specimens after three weeks of g… — ACT

A biology student recorded the mass, in grams, of 6 plant specimens after three weeks of growth: $14$ , $9$ , $22$ , $18$ , $12$ , and $15$ . If a 7th plant specimen is added to the data set, what must its mass be, in grams, so that the mean mass of all 7 specimens equals the median mass of the original 6 specimens?

A
$8.5$
$11.5$ Answer
C
$14.5$
D
$15.0$
E
$50.0$

Answer

The mass of the 7th specimen must be 11.5 grams.

To find the required mass of the 7th specimen, first order the original 6 data points from least to greatest:

9, 12, 14, 15, 18, 22

. The median is the average of the two middle values, 14 and 15, which gives

\frac{14 + 15}{2} = 14.5

grams. The problem states that the mean of all 7 specimens must equal 14.5 grams. The sum of 7 values with a mean of 14.5 is

7 \times 14.5 = 101.5

grams. Since the sum of the first 6 specimens is

9 + 12 + 14 + 15 + 18 + 22 = 90

grams, the mass of the 7th specimen must be

101.5 - 90 = 11.5

grams.

Step-by-Step Solution

Sort the original data set to find its median.

The sorted list of masses is

9, 12, 14, 15, 18, 22

Finding the median of a data set requires placing the numbers in ascending order.

Calculate the median of the original 6 specimens.

\text{Median} = \frac{14 + 15}{2} = 14.5\text{ grams}

For an even number of values, the median is the average of the two middle terms.

Determine the required total mass of all 7 specimens.

\text{Required Total Mass} = 7 \times 14.5 = 101.5\text{ grams}

Mean is calculated by dividing total sum by count, so

\text{Total Sum} = \text{Mean} \times \text{Count}

Calculate the sum of the original 6 specimens.

\text{Original Sum} = 9 + 12 + 14 + 15 + 18 + 22 = 90\text{ grams}

Summing the existing measurements allows us to find the missing amount.

Subtract the original sum from the required total sum.

\text{Mass of 7th Specimen} = 101.5 - 90 = 11.5\text{ grams}

The difference between the required total sum and the current total sum gives the value of the new addition.

Key Concept

Descriptive Statistics: Finding a missing data value given a target mean based on a sample median.

Estimated Time:1m 15s

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