Question
or
Manny has the following data:
2 0 z 6 9 7
If the median is 7, which number could z be?
Answers
GPT-4o mini
To find the potential values for \( z \) such that the median of the set \( \{2, 0, z, 6, 9, 7\} \) equals 7, we must first understand that the median of a sorted dataset is the middle value.
Since there are 6 numbers in the set (an even number), the median will be the average of the 3rd and 4th numbers in the sorted list.
Let's sort the numbers, keeping \( z \) in mind. The current arrangement of fixed numbers (excluding \( z \)) is \( 0, 2, 6, 7, 9 \).
**Case 1: \( z \leq 0 \)**
If \( z \) is less than or equal to 0, the set in sorted order would be \( z, 0, 2, 6, 7, 9 \). In this case, the 3rd and 4th numbers are \( 2 \) and \( 6 \), leading to a median of:
\[
\text{Median} = \frac{2 + 6}{2} = 4
\]
**Case 2: \( 0 < z < 2 \)**
Here, the sorted order is \( 0, z, 2, 6, 7, 9 \). The 3rd and 4th numbers are \( 2 \) and \( 6 \):
\[
\text{Median} = \frac{2 + 6}{2} = 4
\]
**Case 3: \( z = 2 \)**
The sorted set is \( 0, 2, 2, 6, 7, 9 \). The 3rd and 4th numbers are still \( 2 \) and \( 6 \):
\[
\text{Median} = \frac{2 + 6}{2} = 4
\]
**Case 4: \( 2 < z < 6 \)**
In this scenario, the sorted order is \( 0, 2, 6, 7, 9 \). Now, the numbers are still \( 2 \) and \( 6 \):
\[
\text{Median} = \frac{2 + 6}{2} = 4
\]
**Case 5: \( z = 6 \)**
The sorted order is \( 0, 2, 6, 6, 7, 9 \). The 3rd and 4th numbers now are both \( 6 \):
\[
\text{Median} = \frac{6 + 6}{2} = 6
\]
**Case 6: \( 6 < z < 7 \)**
The sorted order is \( 0, 2, 6, 7, 9 \). The 3rd and 4th numbers remain \( 6 \) and \( 7 \):
\[
\text{Median} = \frac{6 + 7}{2} = 6.5
\]
**Case 7: \( z = 7 \)**
Now, the sorted order is \( 0, 2, 6, 7, 7, 9 \). The 3rd and 4th numbers are both \( 7 \):
\[
\text{Median} = \frac{7 + 7}{2} = 7
\]
**Case 8: \( 7 < z < 9 \)**
The sorted set is \( 0, 2, 6, 7, z, 9 \). The 3rd and 4th numbers are \( 6 \) and \( 7 \):
\[
\text{Median} = \frac{6 + 7}{2} = 6.5
\]
**Case 9: \( z \geq 9 \)**
Finally, if \( z \geq 9 \), the sorted order is \( 0, 2, 6, 7, 9, z \). The median is again:
\[
\text{Median} = \frac{6 + 7}{2} = 6.5
\]
### Conclusion
For the median to be exactly \( 7 \), the possible value for \( z \) is:
\[
\boxed{7}
\]
Since there are 6 numbers in the set (an even number), the median will be the average of the 3rd and 4th numbers in the sorted list.
Let's sort the numbers, keeping \( z \) in mind. The current arrangement of fixed numbers (excluding \( z \)) is \( 0, 2, 6, 7, 9 \).
**Case 1: \( z \leq 0 \)**
If \( z \) is less than or equal to 0, the set in sorted order would be \( z, 0, 2, 6, 7, 9 \). In this case, the 3rd and 4th numbers are \( 2 \) and \( 6 \), leading to a median of:
\[
\text{Median} = \frac{2 + 6}{2} = 4
\]
**Case 2: \( 0 < z < 2 \)**
Here, the sorted order is \( 0, z, 2, 6, 7, 9 \). The 3rd and 4th numbers are \( 2 \) and \( 6 \):
\[
\text{Median} = \frac{2 + 6}{2} = 4
\]
**Case 3: \( z = 2 \)**
The sorted set is \( 0, 2, 2, 6, 7, 9 \). The 3rd and 4th numbers are still \( 2 \) and \( 6 \):
\[
\text{Median} = \frac{2 + 6}{2} = 4
\]
**Case 4: \( 2 < z < 6 \)**
In this scenario, the sorted order is \( 0, 2, 6, 7, 9 \). Now, the numbers are still \( 2 \) and \( 6 \):
\[
\text{Median} = \frac{2 + 6}{2} = 4
\]
**Case 5: \( z = 6 \)**
The sorted order is \( 0, 2, 6, 6, 7, 9 \). The 3rd and 4th numbers now are both \( 6 \):
\[
\text{Median} = \frac{6 + 6}{2} = 6
\]
**Case 6: \( 6 < z < 7 \)**
The sorted order is \( 0, 2, 6, 7, 9 \). The 3rd and 4th numbers remain \( 6 \) and \( 7 \):
\[
\text{Median} = \frac{6 + 7}{2} = 6.5
\]
**Case 7: \( z = 7 \)**
Now, the sorted order is \( 0, 2, 6, 7, 7, 9 \). The 3rd and 4th numbers are both \( 7 \):
\[
\text{Median} = \frac{7 + 7}{2} = 7
\]
**Case 8: \( 7 < z < 9 \)**
The sorted set is \( 0, 2, 6, 7, z, 9 \). The 3rd and 4th numbers are \( 6 \) and \( 7 \):
\[
\text{Median} = \frac{6 + 7}{2} = 6.5
\]
**Case 9: \( z \geq 9 \)**
Finally, if \( z \geq 9 \), the sorted order is \( 0, 2, 6, 7, 9, z \). The median is again:
\[
\text{Median} = \frac{6 + 7}{2} = 6.5
\]
### Conclusion
For the median to be exactly \( 7 \), the possible value for \( z \) is:
\[
\boxed{7}
\]