The mean of 15 observation is 23. If each obsevation is multiplied 2, then new mean is .

Options

A. 23
B. 46
C. 25
D. 36

Answer

To solve the problem step by step, we will follow the mathematical concepts related to the mean of observations.

### Step 1: Understand the Given Information
We are given that the mean of 15 observations is 23.

### Step 2: Use the Mean Formula
The formula for the mean (average) is given by:
\[ \text{Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]
Let the sum of all observations be \( S \). Therefore, we can write:
\[ 23 = \frac{S}{15} \]

### Step 3: Calculate the Sum of Observations
To find the sum \( S \), we can rearrange the equation:
\[ S = 23 \times 15 \]
Calculating this gives:
\[ S = 345 \]

### Step 4: Determine the New Observations
Now, if each observation is multiplied by 2, the new observations will be:
\[ 2n_1, 2n_2, 2n_3, \ldots, 2n_{15} \]

### Step 5: Calculate the New Mean
The new mean can be calculated using the new sum of observations:
\[ \text{New Mean} = \frac{2n_1 + 2n_2 + 2n_3 + \ldots + 2n_{15}}{15} \]
This can be simplified by factoring out the 2:
\[ \text{New Mean} = \frac{2(n_1 + n_2 + n_3 + \ldots + n_{15})}{15} \]
Since we already calculated \( n_1 + n_2 + n_3 + \ldots + n_{15} = 345 \), we can substitute this value:
\[ \text{New Mean} = \frac{2 \times 345}{15} \]

### Step 6: Simplify the Expression
Now, we can simplify this expression:
\[ \text{New Mean} = \frac{690}{15} \]
Calculating this gives:
\[ \text{New Mean} = 46 \]

### Conclusion
The new mean after multiplying each observation by 2 is 46.