If the median of `33 ,\ 28 ,\ 20 ,\ 25 ,\ 34 ,\ x\ i s\ 29` , find the maximum possible value of `xdot`

Options

A. 30
B. 31
C. 29
D. 32

Answer

To find the maximum possible value of \( x \) such that the median of the numbers \( 33, 28, 20, 25, 34, x \) is 29, we can follow these steps:

### Step 1: Understand the Median
The median is the middle value of a data set when arranged in ascending order. For an even number of observations, the median is the average of the two middle numbers.

### Step 2: Count the Observations
We have six numbers: \( 33, 28, 20, 25, 34, x \). Since there are 6 numbers (an even count), the median will be the average of the 3rd and 4th observations when arranged in order.

### Step 3: Arrange the Data
To find the median, we need to arrange the numbers in ascending order. The arrangement will depend on the value of \( x \).

### Step 4: Determine the Position of \( x \)
To maximize \( x \), we should place \( x \) in the 4th position when the numbers are sorted. This means that \( x \) should be greater than or equal to the 3rd observation but less than or equal to the 4th observation.

### Step 5: Sort the Known Values
Let's sort the known values \( 20, 25, 28, 33, 34 \):
- If \( x \leq 28 \): The order is \( 20, 25, x, 28, 33, 34 \)
- If \( 28 < x < 33 \): The order is \( 20, 25, 28, x, 33, 34 \)
- If \( x \geq 33 \): The order is \( 20, 25, 28, 33, x, 34 \)

### Step 6: Find the Median
Since we want the median to be 29, we can set up the equation based on the 3rd and 4th observations:
\[ \text{Median} = \frac{\text{3rd observation} + \text{4th observation}}{2} = 29 \]
This implies:
\[ \text{3rd observation} + \text{4th observation} = 58 \]

### Step 7: Calculate the Values
1. If \( x \leq 28 \):
- 3rd observation = \( x \)
- 4th observation = \( 28 \)
- Equation: \( x + 28 = 58 \) → \( x = 30 \) (not possible since \( x \) must be less than or equal to 28)

2. If \( 28 < x < 33 \):
- 3rd observation = \( 28 \)
- 4th observation = \( x \)
- Equation: \( 28 + x = 58 \) → \( x = 30 \) (valid since \( 30 < 33 \))

3. If \( x \geq 33 \):
- 3rd observation = \( 28 \)
- 4th observation = \( 33 \)
- Equation: \( 28 + 33 = 61 \) (not equal to 58)

### Conclusion
The maximum possible value of \( x \) that satisfies the condition is \( 30 \).

### Final Answer
The maximum possible value of \( x \) is \( 30 \).