The mean of a set of observations is `a` . If each observation is multiplied by `b` and each product is decreased by `c`, then the mean of new set of observations is

Options

A. `(a)/(b)+c`
B. `ab-c`
C. `(a)/(b)-c`
D. `ab+c`

Explanation

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the given information
We are given that the mean of a set of observations is \( a \). This means that if we have \( n \) observations \( x_1, x_2, x_3, \ldots, x_n \), the mean can be expressed as:
\[ \text{Mean} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} = a \]

### Step 2: Define the new observations
According to the problem, each observation \( x_i \) is multiplied by \( b \) and then decreased by \( c \). Therefore, the new observations \( y_i \) can be expressed as:
\[ y_i = b \cdot x_i - c \quad \text{for } i = 1, 2, 3, \ldots, n \]

### Step 3: Calculate the mean of the new observations
To find the mean of the new set of observations \( y_1, y_2, \ldots, y_n \), we calculate:
\[ \text{Mean of } y = \frac{y_1 + y_2 + y_3 + \ldots + y_n}{n} \]
Substituting the expression for \( y_i \):
\[ \text{Mean of } y = \frac{(b \cdot x_1 - c) + (b \cdot x_2 - c) + (b \cdot x_3 - c) + \ldots + (b \cdot x_n - c)}{n} \]

### Step 4: Simplify the expression
This can be simplified as follows:
\[ \text{Mean of } y = \frac{b \cdot (x_1 + x_2 + x_3 + \ldots + x_n) - n \cdot c}{n} \]
Using the fact that the sum of the observations \( x_1 + x_2 + x_3 + \ldots + x_n = n \cdot a \):
\[ \text{Mean of } y = \frac{b \cdot (n \cdot a) - n \cdot c}{n} \]

### Step 5: Further simplify
Now we can simplify the expression:
\[ \text{Mean of } y = \frac{b \cdot n \cdot a}{n} - \frac{n \cdot c}{n} \]
This simplifies to:
\[ \text{Mean of } y = b \cdot a - c \]

### Final Result
Thus, the mean of the new set of observations is:
\[ \text{Mean of } y = b \cdot a - c \]