A software quality team tracked the number of bug tickets resolved daily over a period of … — GMAT

A software quality team tracked the number of bug tickets resolved daily over a period of

15

consecutive workdays. The recorded number of resolved tickets for the first

14

days, listed in ascending order, were:

2, 3, 3, 4, 5, 5, 6, 7, 8, 9, 9, 10, 11, 12

If the number of tickets resolved on the

15\text{th}

day was

x

, and the arithmetic mean of the complete

15

-day data set is equal to its median, what is the value of

x

A
$7$
B
$8$
$11$ Cevap
D
$13$
E
$15$

Cevap

The value of

x

11

Summing the initial 14 numbers yields 94. In a set of 15 numbers, the median is the 8th term when ordered. Because there are exactly 7 terms smaller than 7 in the initial list, any value of

x \ge 7

keeps 7 at the 8th position, making the median 7. Setting the mean

\frac{94+x}{15}

equal to 7 gives

94 + x = 105

, which leads to

x = 11

Adım Adım Çözüm

Calculate the sum of the known 14 values.

\text{Sum}_{14} = 2 + 3 + 3 + 4 + 5 + 5 + 6 + 7 + 8 + 9 + 9 + 10 + 11 + 12 = 94

The sum of all 15 terms will be

94 + x

, making the arithmetic mean

\frac{94 + x}{15}

Determine the position of the median in a 15-element set.

The median of an odd-numbered set with

N = 15

is the

\frac{15+1}{2} = 8\text{th}

term when arranged in ascending order.

To set the mean equal to the median, we must determine how

x

affects the 8th term.

Analyze the position of the 8th term based on potential values of

x

The first 14 numbers contain 7 values less than 7 (

2, 3, 3, 4, 5, 5, 6

) and 7 values greater than or equal to 7 (

7, 8, 9, 9, 10, 11, 12

). Therefore, for any

x \ge 7

, the 8th term in the sorted set of 15 numbers is fixed at

7

Since 7 values are strictly less than 7, placing

x \ge 7

ensures that 7 is the 8th smallest value.

Set the arithmetic mean equal to the median (

7

) and solve for

x

\frac{94 + x}{15} = 7 \implies 94 + x = 105 \implies x = 11.

Since

x = 11 \ge 7

, the condition holds and the median remains 7.

Anahtar Kavram

Properties of Mean and Median in a Data Set

Soru

Cevap

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