The mean of 20 observations is 30. If the mean of first 15 observations is 32, find the mean of last 5 observations.

Answer

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

### Step 1: Calculate the sum of all 20 observations
We know that the mean of 20 observations is given as 30. The formula for the mean is:

\[ \text{Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]

Let the sum of all observations be \( S \). Therefore, we can write:

\[ 30 = \frac{S}{20} \]

To find \( S \), we multiply both sides by 20:

\[ S = 30 \times 20 = 600 \]

### Step 2: Calculate the sum of the first 15 observations
We are given that the mean of the first 15 observations is 32. Using the same formula for the mean, we can write:

\[ 32 = \frac{\text{Sum of first 15 observations}}{15} \]

Let the sum of the first 15 observations be \( S_1 \). Therefore:

\[ S_1 = 32 \times 15 = 480 \]

### Step 3: Calculate the sum of the last 5 observations
The last 5 observations can be found by subtracting the sum of the first 15 observations from the sum of all 20 observations:

\[ \text{Sum of last 5 observations} = S - S_1 = 600 - 480 = 120 \]

### Step 4: Calculate the mean of the last 5 observations
Now, we can find the mean of the last 5 observations using the formula for the mean:

\[ \text{Mean of last 5 observations} = \frac{\text{Sum of last 5 observations}}{5} \]

Substituting the sum we found:

\[ \text{Mean of last 5 observations} = \frac{120}{5} = 24 \]

### Final Answer
The mean of the last 5 observations is **24**.

---