The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data, is :
(1) 16.8
(2) 16.0
(3) 15.8
(4) 14.0

Options

A. 16.8
B. `16.0`
C. 15.8
D. `14.0`

Explanation

To solve the problem step by step, we will follow the calculations outlined in the video transcript.

### Step 1: Calculate the initial sum of observations
Given that the mean of the data set comprising 16 observations is 16, we can find the initial sum of the observations using the formula for mean:

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

Substituting the known values:

\[ 16 = \frac{\text{Sum of observations}}{16} \]

To find the sum of observations, we multiply both sides by 16:

\[ \text{Sum of observations} = 16 \times 16 = 256 \]

### Step 2: Adjust the sum after deleting one observation
Next, we need to delete one observation valued at 16. Therefore, we subtract 16 from the initial sum:

\[ \text{New sum after deletion} = 256 - 16 = 240 \]

### Step 3: Add the new observations
Now, we add three new observations valued at 3, 4, and 5. First, we calculate the total of these new observations:

\[ \text{Sum of new observations} = 3 + 4 + 5 = 12 \]

Now, we add this sum to the new sum after deletion:

\[ \text{New sum after addition} = 240 + 12 = 252 \]

### Step 4: Calculate the new number of observations
Initially, there were 16 observations. After deleting one and adding three, the total number of observations becomes:

\[ \text{New number of observations} = 16 - 1 + 3 = 18 \]

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

\[ \text{New mean} = \frac{\text{New sum}}{\text{New number of observations}} = \frac{252}{18} \]

Now we perform the division:

\[ 252 \div 18 = 14 \]

### Conclusion
The new mean of the resultant data set is 14.

### Final Answer
Thus, the correct option is (4) 14.0.

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