Question
Kimberly has the following data:
95 90 y 96 84
If the median is 95, which number could y be?
95 90 y 96 84
If the median is 95, which number could y be?
Answers
GPT-4o mini
To find the possible value of \( y \) such that the median of the dataset \( 95, 90, y, 96, 84 \) is \( 95 \), we should first sort the data, including \( y \), and identify the median position.
The dataset has 5 elements, so the median will be the third number when the data is sorted in ascending order.
1. Start with the known numbers: \( 95, 90, 96, 84 \). Sorting these gives us \( 84, 90, 95, 96 \).
2. There are four numbers, so we consider where \( y \) fits in this order:
- If \( y \leq 84 \): the sorted order will be \( y, 84, 90, 95, 96 \) → Median = 90 (not valid).
- If \( 84 < y \leq 90 \): the sorted order will be \( 84, y, 90, 95, 96 \) → Median = 90 (not valid).
- If \( 90 < y \leq 95 \): the sorted order will be \( 84, 90, y, 95, 96 \) → Median = \( y \) (this can be 95).
- If \( 95 < y < 96 \): the sorted order will be \( 84, 90, 95, y, 96 \) → Median = 95 (valid).
- If \( y = 96 \): the sorted order will be \( 84, 90, 95, 96, 96 \) → Median = 95 (valid).
- If \( y > 96 \): the sorted order will be \( 84, 90, 95, 96, y \) → Median = 95 (valid).
Thus, possible values for \( y \) that keep the median as \( 95 \) are:
- Any value between \( 90 < y < 96 \) (like \( 91, 92, 93, 94, 95 \) but \( 95 \) does not work since then there are two 95s)
- \( y = 96 \)
- Any value greater than \( 96 \)
In summary, \( y \) could be any value in the range \( (90, 96] \), or any number greater than \( 96 \).
The dataset has 5 elements, so the median will be the third number when the data is sorted in ascending order.
1. Start with the known numbers: \( 95, 90, 96, 84 \). Sorting these gives us \( 84, 90, 95, 96 \).
2. There are four numbers, so we consider where \( y \) fits in this order:
- If \( y \leq 84 \): the sorted order will be \( y, 84, 90, 95, 96 \) → Median = 90 (not valid).
- If \( 84 < y \leq 90 \): the sorted order will be \( 84, y, 90, 95, 96 \) → Median = 90 (not valid).
- If \( 90 < y \leq 95 \): the sorted order will be \( 84, 90, y, 95, 96 \) → Median = \( y \) (this can be 95).
- If \( 95 < y < 96 \): the sorted order will be \( 84, 90, 95, y, 96 \) → Median = 95 (valid).
- If \( y = 96 \): the sorted order will be \( 84, 90, 95, 96, 96 \) → Median = 95 (valid).
- If \( y > 96 \): the sorted order will be \( 84, 90, 95, 96, y \) → Median = 95 (valid).
Thus, possible values for \( y \) that keep the median as \( 95 \) are:
- Any value between \( 90 < y < 96 \) (like \( 91, 92, 93, 94, 95 \) but \( 95 \) does not work since then there are two 95s)
- \( y = 96 \)
- Any value greater than \( 96 \)
In summary, \( y \) could be any value in the range \( (90, 96] \), or any number greater than \( 96 \).