Find the resulting mean, if each observation, given above, is :
(i) increased by 3. (ii) decreased by 2.
(iii) multiplied by 4. (iv) divided by 5.

Answer

To solve the question step by step, let's denote the mean of the original observations as \( A \).

### Step 1: Mean of Original Observations
The mean of the original observations \( A_1, A_2, A_3, \ldots, A_n \) can be calculated using the formula:
\[ A = \frac{\sum_{i=1}^{n} A_i}{n} \]

### Step 2: Resulting Mean When Each Observation is Increased by 3
If each observation is increased by 3, the new observations become \( A_1 + 3, A_2 + 3, \ldots, A_n + 3 \). The sum of these new observations is:
\[ \sum_{i=1}^{n} (A_i + 3) = \sum_{i=1}^{n} A_i + 3n \]
Thus, the resulting mean is:
\[ \text{Resulting Mean} = \frac{\sum_{i=1}^{n} A_i + 3n}{n} = \frac{\sum_{i=1}^{n} A_i}{n} + \frac{3n}{n} = A + 3 \]

### Step 3: Resulting Mean When Each Observation is Decreased by 2
If each observation is decreased by 2, the new observations become \( A_1 - 2, A_2 - 2, \ldots, A_n - 2 \). The sum of these new observations is:
\[ \sum_{i=1}^{n} (A_i - 2) = \sum_{i=1}^{n} A_i - 2n \]
Thus, the resulting mean is:
\[ \text{Resulting Mean} = \frac{\sum_{i=1}^{n} A_i - 2n}{n} = \frac{\sum_{i=1}^{n} A_i}{n} - \frac{2n}{n} = A - 2 \]

### Step 4: Resulting Mean When Each Observation is Multiplied by 4
If each observation is multiplied by 4, the new observations become \( 4A_1, 4A_2, \ldots, 4A_n \). The sum of these new observations is:
\[ \sum_{i=1}^{n} (4A_i) = 4 \sum_{i=1}^{n} A_i \]
Thus, the resulting mean is:
\[ \text{Resulting Mean} = \frac{4 \sum_{i=1}^{n} A_i}{n} = 4 \cdot \frac{\sum_{i=1}^{n} A_i}{n} = 4A \]

### Step 5: Resulting Mean When Each Observation is Divided by 5
If each observation is divided by 5, the new observations become \( \frac{A_1}{5}, \frac{A_2}{5}, \ldots, \frac{A_n}{5} \). The sum of these new observations is:
\[ \sum_{i=1}^{n} \left(\frac{A_i}{5}\right) = \frac{1}{5} \sum_{i=1}^{n} A_i \]
Thus, the resulting mean is:
\[ \text{Resulting Mean} = \frac{\frac{1}{5} \sum_{i=1}^{n} A_i}{n} = \frac{1}{5} \cdot \frac{\sum_{i=1}^{n} A_i}{n} = \frac{A}{5} \]

### Summary of Results
1. If each observation is increased by 3, the resulting mean is \( A + 3 \).
2. If each observation is decreased by 2, the resulting mean is \( A - 2 \).
3. If each observation is multiplied by 4, the resulting mean is \( 4A \).
4. If each observation is divided by 5, the resulting mean is \( \frac{A}{5} \).