The mean of 15 observations is 30. Two observations 28 and 38 are deleted and three observations 33, 39, and 48 are included. Find the mean of new set of observations.

Options

A. 31
B. `31.5`
C. 32
D. `33.4`

Answer

To find the mean of the new set of observations after deleting two observations and adding three new ones, we can follow these steps:

### Step 1: Calculate the sum of the original observations
We know that the mean of 15 observations is 30. The formula for the mean is:

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

From this, we can find the sum of the original observations:

\[ \text{Sum of observations} = \text{Mean} \times \text{Number of observations} = 30 \times 15 = 450 \]

### Step 2: Delete the two observations
We need to delete the observations 28 and 38 from the sum of observations:

\[ \text{New sum after deletion} = 450 - 28 - 38 \]

Calculating this gives:

\[ \text{New sum after deletion} = 450 - 66 = 384 \]

### Step 3: Add the three new observations
Now we add the new observations 33, 39, and 48 to the new sum:

\[ \text{New sum after addition} = 384 + 33 + 39 + 48 \]

Calculating this gives:

\[ \text{New sum after addition} = 384 + 120 = 504 \]

### Step 4: Calculate the new number of observations
Initially, we had 15 observations. After deleting 2 and adding 3, the new number of observations is:

\[ \text{New number of observations} = 15 - 2 + 3 = 16 \]

### Step 5: Calculate the new mean
Finally, we can find the new mean using the new sum of observations and the new number of observations:

\[ \text{New Mean} = \frac{\text{New sum}}{\text{New number of observations}} = \frac{504}{16} \]

Calculating this gives:

\[ \text{New Mean} = 31.5 \]

### Final Answer
The mean of the new set of observations is **31.5**.

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