The standard deviation of 25 observations is 4 and their mean is 25. If each observation is increased by 10, what is the new mean-

Options

A. (a) 25
B. (b) 29
C. c. 30
D. (d) 35.

Answer

To solve the problem step by step, we need to find the new mean after increasing each observation by 10.

### Step 1: Understand the given information
We are given:
- The mean of the original 25 observations (let's denote it as \( \bar{x} \)) is 25.
- The number of observations (n) is 25.
- Each observation is increased by 10.

### Step 2: Write the formula for the mean
The mean of a set of observations is calculated using the formula:
\[ \bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n} \]
where \( x_1, x_2, \ldots, x_n \) are the observations.

### Step 3: Calculate the new observations
If each observation \( x_i \) is increased by 10, the new observations can be expressed as:
\[ x_1 + 10, x_2 + 10, \ldots, x_n + 10 \]
Thus, the new observations can be represented as:
\[ \text{New observations} = (x_1 + 10), (x_2 + 10), \ldots, (x_{25} + 10) \]

### Step 4: Calculate the new mean
The new mean \( \bar{x}_{new} \) can be calculated as follows:
\[ \bar{x}_{new} = \frac{(x_1 + 10) + (x_2 + 10) + \ldots + (x_{25} + 10)}{25} \]
This can be simplified:
\[ \bar{x}_{new} = \frac{(x_1 + x_2 + \ldots + x_{25}) + (10 \times 25)}{25} \]
Since we know that \( \bar{x} = \frac{x_1 + x_2 + \ldots + x_{25}}{25} = 25 \), we can substitute:
\[ \bar{x}_{new} = \frac{(25 \times 25) + 250}{25} \]
Calculating the numerator:
\[ \bar{x}_{new} = \frac{625 + 250}{25} = \frac{875}{25} = 35 \]

### Conclusion
The new mean after increasing each observation by 10 is **35**.

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