The median of a set of nine distinct observations is 20.5. If each of the last four observations of the set is increased by 2, then the median of the new set is

Options

A. Is increased by 2
B. In decreased by 2
C. Is two times the original median
D. Remains the same as that of the original set

Explanation

To solve the problem step by step, we need to analyze the information given about the observations and how the median is affected by the changes made to the last four observations.

### 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 9 distinct observations, the median is the 5th observation when arranged in order.

### Step 2: Identify the Observations
Let’s denote the observations as \( x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9 \). Given that the median is 20.5, we know that:
\[ x_5 = 20.5 \]
This means that when the observations are arranged in ascending order, the 5th observation is 20.5.

### Step 3: Modify the Last Four Observations
According to the problem, the last four observations \( x_6, x_7, x_8, x_9 \) are each increased by 2. Therefore, the new values will be:
\[ x_6 + 2, \quad x_7 + 2, \quad x_8 + 2, \quad x_9 + 2 \]

### Step 4: Determine the New Median
Since the median is the 5th observation in the ordered list, we need to check if the 5th observation changes after modifying the last four observations. The 5th observation \( x_5 \) remains unchanged at 20.5 because it is not affected by the changes made to \( x_6, x_7, x_8, \) and \( x_9 \).

### Step 5: Conclusion
Since the 5th observation (the median) remains unchanged, the median of the new set after increasing the last four observations by 2 is still:
\[ \text{Median} = 20.5 \]

### Final Answer
Thus, the median of the new set is **20.5**.

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