The mean of twenty observations is 15. If two observations 3 and 14 are replaced by 8 and 9 respectively, then what will be the new mean.

Answer

To solve the problem step by step, we will follow the process of calculating the new mean after replacing the specified observations.

### Step 1: Calculate the sum of the original observations
The mean of the original 20 observations is given as 15. The formula for mean is:

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

Given that the mean is 15 and the number of observations is 20, we can find the sum of the observations:

\[ \text{Sum of observations} = \text{Mean} \times \text{Number of observations} = 15 \times 20 = 300 \]

### Step 2: Identify the observations to be replaced
We need to replace two observations: 3 and 14.

### Step 3: Calculate the new sum after replacing the observations
To find the new sum of observations after replacing 3 and 14 with 8 and 9, we first need to subtract the old observations and add the new ones:

\[ \text{New Sum} = \text{Old Sum} - (\text{Old Observations}) + (\text{New Observations}) \]

Substituting the values:

\[ \text{New Sum} = 300 - (3 + 14) + (8 + 9) \]

Calculating the values inside the parentheses:

\[ \text{New Sum} = 300 - 17 + 17 \]

This simplifies to:

\[ \text{New Sum} = 300 \]

### Step 4: Calculate the new mean
Now that we have the new sum of observations, we can calculate the new mean using the same number of observations (20):

\[ \text{New Mean} = \frac{\text{New Sum}}{\text{Number of observations}} = \frac{300}{20} = 15 \]

### Conclusion
The new mean after replacing the observations is **15**.

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