Answer :
Let's analyze the effect of adding a constant to each observation in a set on the mean of the set.
Assume we have a set \( S \) with \( n \) observations: \( S = \{x_1, x_2, x_3, \ldots, x_n\} \).
The mean \( \bar{x} \) of the original set \( S \) is calculated as the sum of all the observations divided by the number of observations:
\[ \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i \]
Now, let's create a new set \( S' \) where we add 5 to each observation from the original set \( S \): \( S' = \{x_1 + 5, x_2 + 5, x_3 + 5, \ldots, x_n + 5\} \).
The mean \( \bar{x'} \) of the new set \( S' \) is calculated similarly:
\[ \bar{x'} = \frac{1}{n} \sum_{i=1}^n (x_i + 5) \]
\[ \bar{x'} = \frac{1}{n} \left( \sum_{i=1}^n x_i + \sum_{i=1}^n 5 \right) \]
\[ \bar{x'} = \frac{1}{n} \sum_{i=1}^n x_i + \frac{1}{n} \cdot n \cdot 5 \]
\[ \bar{x'} = \bar{x} + 5 \]
Thus, when we add a constant 5 to each observation in the set, the mean of the set is increased by the same constant, which is 5. This concept is true for any constant and any original set of data.
So, the correct answer to the question is:
(a) increased by 5
