The mean of 15 observations is 32. Find the resulting mean, if each observation is increased by 3

Answer

To solve the problem step by step, we will follow the given information and apply the concepts of mean.

### Step 1: Understand the given information
We know that the mean of 15 observations is 32.

### Step 2: Calculate the sum of the observations
The formula for mean is:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Total number of observations}} \]
Given that the mean is 32 and the total number of observations is 15, we can write:
\[ 32 = \frac{\text{Sum of observations}}{15} \]
To find the sum of observations, we multiply both sides by 15:
\[ \text{Sum of observations} = 32 \times 15 = 480 \]

### Step 3: Determine the new sum after increasing each observation by 3
If each observation is increased by 3, then the total increase in the sum will be:
\[ \text{Total increase} = 3 \times \text{Total number of observations} = 3 \times 15 = 45 \]
Now, we can find the new sum of observations:
\[ \text{New sum} = \text{Old sum} + \text{Total increase} = 480 + 45 = 525 \]

### Step 4: Calculate the new mean
Now that we have the new sum of observations, we can find the new mean using the same formula:
\[ \text{New Mean} = \frac{\text{New sum}}{\text{Total number of observations}} = \frac{525}{15} \]
Calculating this gives:
\[ \text{New Mean} = 35 \]

### Final Answer
Thus, the resulting mean after increasing each observation by 3 is **35**.

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