The mean of 17 observations is 30, three observations 20, 30, and 40 are deleted and one observation 90 is included. Then, find the mean of the final observations.

Options

A. 33
B. 34
C. 35
D. 36

Answer

To find the mean of the final observations after deleting three observations and adding one, we can follow these steps:

### Step 1: Calculate the sum of the original observations
Given that the mean of 17 observations is 30, we can find the total sum of these observations using the formula for mean:

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

Let \( S \) be the sum of the observations. Therefore:

\[ 30 = \frac{S}{17} \]

Multiplying both sides by 17 gives:

\[ S = 30 \times 17 = 510 \]

### Step 2: Subtract the sum of the deleted observations
We need to delete three observations: 20, 30, and 40. First, we calculate their total:

\[ \text{Sum of deleted observations} = 20 + 30 + 40 = 90 \]

Now, we subtract this sum from the original sum:

\[ \text{New sum after deletion} = 510 - 90 = 420 \]

### Step 3: Add the new observation
Next, we add the new observation, which is 90:

\[ \text{Final sum} = 420 + 90 = 510 \]

### Step 4: Calculate the number of final observations
After deleting three observations and adding one, the number of observations becomes:

\[ \text{Final number of observations} = 17 - 3 + 1 = 15 \]

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

\[ \text{Final Mean} = \frac{\text{Final sum}}{\text{Final number of observations}} = \frac{510}{15} \]

Calculating this gives:

\[ \text{Final Mean} = 34 \]

### Final Answer
The mean of the final observations is **34**.

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