The mean of 15 observations is 32. Find the resulting mean if each observation is :
decreased by 20%

Answer

To find the resulting mean when each observation is decreased by 20%, we can follow these steps:

### Step 1: Understand the given information
We know that the mean of 15 observations is 32. This means that the sum of all observations can be calculated as:
\[ \text{Sum of observations} = \text{Mean} \times \text{Number of observations} = 32 \times 15 \]

### Step 2: Calculate the sum of the observations
Calculating the sum:
\[ \text{Sum of observations} = 32 \times 15 = 480 \]

### Step 3: Determine the effect of decreasing each observation by 20%
Decreasing each observation by 20% means that each observation will now be 80% of its original value. If we denote the original observations as \( x_1, x_2, \ldots, x_{15} \), the new observations will be:
\[ 0.8x_1, 0.8x_2, \ldots, 0.8x_{15} \]

### Step 4: Calculate the new sum of the observations
The new sum of the observations can be expressed as:
\[ \text{New Sum} = 0.8(x_1 + x_2 + \ldots + x_{15}) = 0.8 \times 480 \]

### Step 5: Perform the multiplication
Calculating the new sum:
\[ \text{New Sum} = 0.8 \times 480 = 384 \]

### Step 6: Calculate the new mean
To find the new mean, we divide the new sum by the number of observations:
\[ \text{New Mean} = \frac{\text{New Sum}}{\text{Number of observations}} = \frac{384}{15} \]

### Step 7: Perform the division
Calculating the new mean:
\[ \text{New Mean} = \frac{384}{15} = 25.6 \]

### Final Result
Thus, the resulting mean when each observation is decreased by 20% is:
\[ \text{New Mean} = 25.6 \]

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