Let `barx` be The the mean of `x_(1),x_(2),x_(3)…x_(n)` if `x_(i) = a +cy_(i)` , for some constants a and c. then what will be the mean of `y_(1),y_(2),y_(3)…y_(n)`

Options

A. `a+barcx`
B. `a-(1)/(c )barc`
C. `(1)/(c )barx`
D. `(barx-a)/(c )`

Answer

To solve the problem, we need to find the mean of the series \( y_1, y_2, y_3, \ldots, y_n \) given that \( x_i = a + c y_i \) for some constants \( a \) and \( c \).

### Step-by-step Solution:

1. **Understand the Given Information**:
We are given that \( x_i = a + c y_i \). This means each \( x_i \) is derived from \( y_i \) through a linear transformation involving constants \( a \) and \( c \).

2. **Calculate the Mean of \( x_i \)**:
The mean of the \( x_i \) values is given by:
\[ \bar{x} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \]

3. **Substitute the Expression for \( x_i \)**:
Substitute \( x_i = a + c y_i \) into the mean formula:
\[ \bar{x} = \frac{(a + c y_1) + (a + c y_2) + (a + c y_3) + \ldots + (a + c y_n)}{n} \]

4. **Simplify the Expression**:
This can be simplified as follows:
\[ \bar{x} = \frac{na + c(y_1 + y_2 + y_3 + \ldots + y_n)}{n} \]
\[ \bar{x} = a + \frac{c(y_1 + y_2 + y_3 + \ldots + y_n)}{n} \]

5. **Express the Mean of \( y_i \)**:
Let \( \bar{y} \) be the mean of \( y_1, y_2, y_3, \ldots, y_n \):
\[ \bar{y} = \frac{y_1 + y_2 + y_3 + \ldots + y_n}{n} \]

6. **Relate \( \bar{x} \) and \( \bar{y} \)**:
From the previous step, we can express \( \bar{y} \) in terms of \( \bar{x} \):
\[ \bar{x} = a + c \bar{y} \]

7. **Solve for \( \bar{y} \)**:
Rearranging the equation gives:
\[ c \bar{y} = \bar{x} - a \]
\[ \bar{y} = \frac{\bar{x} - a}{c} \]

### Final Result:
Thus, the mean of \( y_1, y_2, y_3, \ldots, y_n \) is:
\[ \bar{y} = \frac{\bar{x} - a}{c} \]