A researcher records the daily high temperatures, in degrees Fahrenheit, for 9 consecutive… — SAT

A researcher records the daily high temperatures, in degrees Fahrenheit, for 9 consecutive days in a certain city: $72$ , $74$ , $75$ , $77$ , $78$ , $79$ , $81$ , $82$ , and $83$ . On the 10th day, the temperature was $x$ degrees Fahrenheit. If the median of the 10-day dataset is $78.5$ degrees Fahrenheit, which of the following could be the value of $x$ ?

A
75
B
76
C
78
80Answer

Answer

To find the median of a dataset with 10 values, we sort the values in ascending order and average the 5th and 6th values. The original 9 values in sorted order are

72, 74, 75, 77, 78, 79, 81, 82, 83

. For the median of the 10-value dataset to be

78.5

, the 5th value must be

78

and the 6th value must be

79

. If the new value

x

is greater than or equal to

79

, the first 5 values will remain

72, 74, 75, 77, 78

, and the 6th value will be

79

. This results in a median of

\frac{78 + 79}{2} = 78.5

. Since

80

is greater than or equal to

79

, it could be the value of

x

Step-by-Step Solution

Determine the rule for the median of a dataset with 10 values.

The median of 10 values is the average of the 5th and 6th values when the dataset is sorted in ascending order.

This is the definition of the median for an even number of data points.

Set up an equation for the 5th and 6th values.

For the median to be

78.5

, the 5th and 6th values in the sorted list must average to

78.5

. Since all temperatures are integers, the 5th value must be

78

and the 6th value must be

79

The average of

78

and

79

\frac{78 + 79}{2} = 78.5

Find the inequality restriction on

x

The original sorted list has 5 values less than or equal to

78

72, 74, 75, 77, 78

. If the 10th value

x

is greater than or equal to

79

, it will be placed in position 6 or higher. The first 5 values will remain in positions 1 to 5, meaning the 5th value is

78

and the 6th value is

79

. Thus,

x

must be greater than or equal to

79

x

were less than

79

, it would shift

78

to the 6th position, resulting in a median that is less than or equal to

78

Identify the option that satisfies the restriction.

Among the options, only

80

is greater than or equal to

79

A value of

80

for

x

ensures the sorted list is

72, 74, 75, 77, 78, 79, 80, 81, 82, 83

, which has a median of

78.5

Key Concept

Calculating the median of a dataset with an even number of elements and understanding how the addition of a new element affects the positions of sorted values.

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Key Concept

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