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 will analyze the given information and apply the properties of the median.

### 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 with an odd number of observations, the median is the value at the position \((n + 1) / 2\), where \(n\) is the number of observations.

### Step 2: Identify the Position of the Median
In this case, we have 9 distinct observations. Therefore, the median is located at the position:
\[ \text{Median Position} = \frac{9 + 1}{2} = 5 \]
This means that the 5th observation in the ordered set is the median.

### Step 3: Given Median Value
We are given that the median (5th observation) is 20.5. So, we can denote the ordered observations as:
\[ x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9 \]
where \(x_5 = 20.5\).

### Step 4: 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 observations will be:
\[ x_6 + 2, x_7 + 2, x_8 + 2, x_9 + 2 \]

### Step 5: Analyze the Effect on the Median
The median is determined by the middle value of the ordered set. Since we are only increasing the last four observations, the first five observations \(x_1, x_2, x_3, x_4, x_5\) remain unchanged. The new ordered set will look like this:
\[ x_1, x_2, x_3, x_4, x_5, (x_6 + 2), (x_7 + 2), (x_8 + 2), (x_9 + 2) \]
Since \(x_5\) is still the 5th observation and remains unchanged at 20.5, the median of the new set will still be the 5th observation.

### Step 6: Conclusion
Thus, the median of the new set after increasing the last four observations by 2 remains:
\[ \text{New Median} = 20.5 \]

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