The mean of 1,7,5,3,4 and 4 is m. The observations 3,2,4,2,3,3 and p have the mean (m-1) Find the median of this set of data.

Options

A. 4
B. `2.5`
C. 3
D. 5

Answer

To solve the problem step by step, we will follow these procedures:

### Step 1: Calculate the mean \( m \) of the first set of numbers
The first set of numbers is: 1, 7, 5, 3, 4, and 4.

**Calculation:**
- Sum of the numbers = \( 1 + 7 + 5 + 3 + 4 + 4 = 24 \)
- Number of observations = 6
- Mean \( m = \frac{\text{Sum of observations}}{\text{Number of observations}} = \frac{24}{6} = 4 \)

### Step 2: Determine the mean of the second set of numbers
The second set of numbers is: 3, 2, 4, 2, 3, 3, and \( p \). We know that the mean of this set is \( m - 1 \).

**Calculation:**
- Since \( m = 4 \), then \( m - 1 = 4 - 1 = 3 \).
- Mean of the second set = 3
- Therefore, \( 3 = \frac{3 + 2 + 4 + 2 + 3 + 3 + p}{7} \)

### Step 3: Solve for \( p \)
**Calculation:**
- Sum of the known numbers = \( 3 + 2 + 4 + 2 + 3 + 3 = 17 \)
- Setting up the equation:
\[ 3 = \frac{17 + p}{7} \]
- Multiply both sides by 7:
\[ 21 = 17 + p \]
- Solve for \( p \):
\[ p = 21 - 17 = 4 \]

### Step 4: List the complete second set of numbers
Now we have the complete second set of numbers: 3, 2, 4, 2, 3, 3, and \( p = 4 \).

### Step 5: Arrange the second set in ascending order
The numbers in ascending order are: 2, 2, 3, 3, 3, 4, 4.

### Step 6: Find the median of the second set
Since there are 7 observations (an odd number), the median is the middle value.

**Calculation:**
- The median position is given by \( \frac{n + 1}{2} \) where \( n \) is the number of observations.
- Here, \( n = 7 \):
\[ \text{Median position} = \frac{7 + 1}{2} = 4 \]
- The 4th value in the ordered list (2, 2, 3, 3, 3, 4, 4) is 3.

### Final Answer
Thus, the median of the second set of data is **3**.

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