To solve the problem, we need to find the median of the observations given that the mode is 3. Let's go through the steps one by one.
### Step 1: Identify the Observations
The observations given are:
\[ 5, 4, 4, 3, 5, x, 3, 4, 3, 5, 4, 3, 5 \]
### Step 2: Determine the Value of \( x \)
We know that the mode of the observations is 3. The mode is the number that appears most frequently in a data set.
Let's count the occurrences of each number:
- The number of 5s: 5 appears 4 times.
- The number of 4s: 4 appears 4 times.
- The number of 3s: 3 appears 4 times.
- The number of \( x \): We need to determine \( x \).
Since the mode is 3, it must occur more frequently than any other number. Therefore, to maintain 3 as the mode, \( x \) must be equal to 3.
Thus, we have:
\[ x = 3 \]
### Step 3: Rewrite the Observations
Now we can rewrite the observations with \( x \) replaced:
\[ 5, 4, 4, 3, 5, 3, 3, 4, 3, 5, 4, 3, 5 \]
### Step 4: Arrange the Observations in Ascending Order
Next, we need to arrange these observations in ascending order:
\[ 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5 \]
### Step 5: Count the Total Number of Observations
Now, let's count the total number of observations:
- Total observations = 13
### Step 6: Calculate the Median
Since the total number of observations (n) is odd, we can find the median using the formula:
\[ \text{Median} = \text{the } \left(\frac{n + 1}{2}\right)^{th} \text{ element} \]
Substituting \( n = 13 \):
\[ \text{Median} = \left(\frac{13 + 1}{2}\right)^{th} \text{ element} = \left(\frac{14}{2}\right)^{th} \text{ element} = 7^{th} \text{ element} \]
### Step 7: Identify the 7th Element
Now, we find the 7th element in the ordered list:
- The ordered list is: \( 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5 \)
- The 7th element is 4.
### Final Answer
Thus, the median of the observations is:
\[ \text{Median} = 4 \]
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