An environmental monitoring station recorded the daily nitrogen dioxide (\text{NO}_2) conc… — GRE General Test

An environmental monitoring station recorded the daily nitrogen dioxide ( $\text{NO}_2$ ) concentrations, in parts per billion (ppb), for 6 consecutive days: $19, 28, 30, 35, 42,$ and $42$ . On the 7th day, a measurement of $x$ ppb was recorded, where $x > 45$ . If the arithmetic mean of the 7 daily concentrations is equal to their median, what is the value of $x$ ?

Cevap: 49 ppb

Cevap

The value of

x

49

Sorting the first 6 recorded concentrations gives

19, 28, 30, 35, 42, 42

. Because

x > 45

x

is strictly greater than all existing values, so the full set in ascending order is

19, 28, 30, 35, 42, 42, x

. The median of a 7-element set is the 4th value, which is

35

. The sum of the 7 concentrations is

19 + 28 + 30 + 35 + 42 + 42 + x = 196 + x

, making the arithmetic mean

\frac{196 + x}{7}

. Setting the mean equal to the median gives

\frac{196 + x}{7} = 35 \implies 196 + x = 245 \implies x = 49

Adım Adım Çözüm

Order the first 6 data points in ascending order

The sorted list is

19, 28, 30, 35, 42, 42

Establishing ordered positions is necessary to determine the median.

Determine the median of the 7-element dataset

Since

x > 45

, the complete ordered dataset is

19, 28, 30, 35, 42, 42, x

, making the 4th element,

35

, the median.

The median of an odd number of sorted values is the middle element.

Express the arithmetic mean in terms of

x

and set it equal to the median

\frac{196 + x}{7} = 35

The problem states that the arithmetic mean equals the median.

Solve the linear equation for

x

196 + x = 245 \implies x = 49

Multiplying both sides by 7 and subtracting 196 isolates

x

Anahtar Kavram

Measures of Central Tendency (Mean and Median)

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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