The median of a set of 11 distinct observations is `15.5`. If each of the smallest 5 observations of the set are decreased by 3, then the median of the new set :

Options

A. In increased by 3
B. Is decreased by 3
C. Is three times the original median
D. Remains the same as that of the original set

Answer

To solve the problem step by step, we will analyze the given information and apply the concept of median in a systematic way.

### Step 1: Understand the Median
The median of a set of observations is the middle value when the observations are arranged in ascending order. For a set of 11 observations, the median will be the 6th observation when arranged in order.

**Hint:** Remember that the median is the middle value in an ordered list of numbers.

### Step 2: Identify the Current Median
We are given that the median of the original set of 11 distinct observations is `15.5`. This means that the 6th observation (when arranged in ascending order) is `15.5`.

**Hint:** The position of the median in a set of 11 numbers is calculated as (n + 1)/2, where n is the number of observations.

### Step 3: Analyze the Changes to the Observations
The problem states that the smallest 5 observations are decreased by 3. Let’s denote the smallest 5 observations as \( x_1, x_2, x_3, x_4, x_5 \). After decreasing each by 3, they become \( x_1 - 3, x_2 - 3, x_3 - 3, x_4 - 3, x_5 - 3 \).

**Hint:** Focus on how the changes affect the order of the observations, especially around the median position.

### Step 4: Determine the New Median
Since we only decreased the smallest 5 observations, the 6th observation (which is the median) remains unchanged because it is not affected by the changes made to the smallest 5 observations. The 6th observation is still `15.5`.

**Hint:** Check if the position of the median (6th term) is affected by the changes made to the other observations.

### Step 5: Conclusion
Since the 6th observation remains the same, the new median of the modified set of observations will also be `15.5`.

**Final Answer:** The median of the new set is `15.5`.