Question

tristan records the number of customers who visit the store each hour on a saturday. his data representing the first seven hours are 15, 23, 12, 28, 20, 18, and 23. how many customers visited the store during the eighth hour if the median number of customers per hour did not change?
○ 16
○ 19
○ 20
○ 23

Explanation:

Step1: Order the initial data

First, we order the initial seven data points: 12, 15, 18, 20, 23, 23, 28.
For a set of seven numbers (odd number of elements), the median is the middle number, which is the 4th number. So the median of the initial set is 20.

Step2: Consider the eighth data point

When we add an eighth data point, the set will have 8 elements (even number of elements). The median of a set with 8 elements is the average of the 4th and 5th numbers when the data is ordered. We want the median to remain 20. Let the eighth data point be \( x \). We order the eight data points. The first seven ordered are 12, 15, 18, 20, 23, 23, 28, and \( x \) will be inserted into the correct position.

To have the median (average of 4th and 5th) equal to 20, we need \(\frac{\text{4th number}+\text{5th number}}{2}=20\). The 4th number in the eight - element set should be 20 (to keep the median the same as before when we consider the average). So when we insert \( x \), the 4th number is 20 and the 5th number should also be 20 (so that \(\frac{20 + 20}{2}=20\)). This means that \( x\) must be less than or equal to 20 (to keep the 4th number as 20) and when we order the eight numbers, the 5th number is 20. If we take \( x = 20\), the ordered eight numbers are 12, 15, 18, 20, 20, 23, 23, 28. The median is \(\frac{20+20}{2}=20\), which is the same as the original median.

Let's check other options:

• If \( x = 16\): Ordered data: 12, 15, 16, 18, 20, 23, 23, 28. Median is \(\frac{18 + 20}{2}=19

eq20\).

• If \( x = 19\): Ordered data: 12, 15, 18, 19, 20, 23, 23, 28. Median is \(\frac{19+20}{2}=19.5

eq20\).

• If \( x = 23\): Ordered data: 12, 15, 18, 20, 23, 23, 23, 28. Median is \(\frac{20 + 23}{2}=21.5

eq20\).

Answer:

20 (corresponding to the option "20")