A researcher records the following list of seven temperatures (in degrees Fahrenheit) duri… — ACT

A researcher records the following list of seven temperatures (in degrees Fahrenheit) during a week, where $x$ and $y$ are unknown values:

88, 92, 75, 85, x, y, 90

The mean temperature for the seven days is $85.0^\circ\text{F}$ , and the range of the temperatures is $22.0^\circ\text{F}$ . If the highest temperature of the week is $y$ and the lowest temperature of the week is $x$ , what is the median temperature of the week?

A
$82.5^\circ\text{F}$
B
$85.0^\circ\text{F}$
C
$86.5^\circ\text{F}$
$88.0^\circ\text{F}$ Cevap
E
$90.0^\circ\text{F}$

Cevap

88.0^\circ\text{F}

To find the median temperature, we must first find the values of

x

and

y

. Since the mean of the seven temperatures is

85.0^\circ\text{F}

, their sum is

7 \times 85.0 = 595.0

. The sum of the five known temperatures is

88 + 92 + 75 + 85 + 90 = 430.0

. Therefore, the sum of the two unknown temperatures is

x + y = 595.0 - 430.0 = 165.0

. We are also given that the range of the temperatures is

22.0^\circ\text{F}

, and that

y

is the highest and

x

is the lowest temperature, which means

y - x = 22.0

. Solving this system of equations (

y + x = 165.0

and

y - x = 22.0

) yields

x = 71.5

and

y = 93.5

. Placing the seven temperatures in ascending order gives

71.5, 75, 85, 88, 90, 92, 93.5

. The median is the middle value (the 4th value) in this sorted list, which is

88.0^\circ\text{F}

Adım Adım Çözüm

Determine the sum of the temperatures using the given mean of

85.0^\circ\text{F}

Total sum =

595.0

Since the mean of 7 temperatures is

85.0

, the sum of all temperatures must be

7 \times 85.0 = 595.0

Set up an equation for the sum of the unknown temperatures

x

and

y

x + y = 165.0

The sum of the five known temperatures is

88 + 92 + 75 + 85 + 90 = 430.0

. Thus,

x + y = 595.0 - 430.0 = 165.0

Set up and solve the system of equations with the range constraint to find

x

and

y

x = 71.5

and

y = 93.5

We are given that

y

is the maximum and

x

is the minimum, so the range is

y - x = 22.0

. Solving the system

y + x = 165.0

and

y - x = 22.0

by adding the equations gives

2y = 187.0 \implies y = 93.5

. Substituting back gives

x = 71.5

Sort the seven temperatures in ascending order and identify the median value.

Sorted list:

71.5, 75, 85, 88, 90, 92, 93.5

; Median =

88.0

For an odd number of data points (7), the median is the 4th value when the list is sorted. The 4th value in the sorted list is

88.0

Anahtar Kavram

Calculating the median of a dataset containing unknown values by deriving and solving a system of linear equations based on the mean and the range.

Alternatif Yöntem

Instead of setting up and solving the system of equations algebraically, one can test the median by using the fact that the sum of the deviations from the mean (

85

) must equal

0

. The deviations of the known numbers from

85

are:

(88-85) + (92-85) + (75-85) + (85-85) + (90-85) = 3 + 7 - 10 + 0 + 5 = 5

. Therefore, the sum of the deviations of

x

and

y

from

85

must be

-5

(x-85) + (y-85) = -5

, which simplifies to

x + y - 170 = -5

, or

x + y = 165

. Since

y - x = 22

, we can quickly find

x = 71.5

and

y = 93.5

, then sort the list to find the median.

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Cevap

Adım Adım Çözüm

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