A coffee shop owner records the number of customers served during each hour of an 8-hour s… — SAT

A coffee shop owner records the number of customers served during each hour of an 8-hour shift. The recorded numbers of customers are $11$ , $13$ , $14$ , $15$ , $17$ , $18$ , $20$ , and $x$ . If the mean number of customers served per hour is equal to the median number of customers served per hour for this shift, and $x > 18$ , what is the value of $x$ ?

Cevap: 20

Cevap

The correct value is 20. When

x > 18

, the 4th and 5th values of the sorted dataset are 15 and 17, giving a median of 16. Setting the mean, which is (108 + x)/8, equal to 16 yields x = 20, which is consistent with the condition

x > 18

Adım Adım Çözüm

Determine the median of the dataset given the constraint

x > 18

The median of the dataset is

16

When the 8 values are sorted in ascending order, the first six values must be 11, 13, 14, 15, 17, and 18. The remaining two values, 20 and

x

, will occupy the 7th and 8th positions (in either order). Since there are 8 values, the median is the average of the 4th and 5th values:

\frac{15 + 17}{2} = 16

Set up the equation for the mean of the dataset.

The mean is represented by the expression

\frac{108 + x}{8}

The mean is the sum of all 8 values divided by 8:

\frac{11 + 13 + 14 + 15 + 17 + 18 + 20 + x}{8} = \frac{108 + x}{8}

Equate the mean to the median and solve for

x

x = 20

Setting the mean equal to the median gives

\frac{108 + x}{8} = 16

. Multiplying by 8 yields

108 + x = 128

. Subtracting 108 from both sides results in

x = 20

, which satisfies the condition

x > 18

Anahtar Kavram

Analyzing the mean and median of a dataset containing a variable constraint

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