The mean of 15 observations is 32. Find the resulting mean, if each observation is decreased by 7

Answer

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

### Step 1: Understand the given information
We know that the mean of 15 observations is 32.

### Step 2: Calculate the sum of the observations
The mean is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Given that the mean is 32 and the number of observations is 15, we can find the sum of the observations:
\[ 32 = \frac{\text{Sum of observations}}{15} \]
To find the sum of observations, we multiply both sides by 15:
\[ \text{Sum of observations} = 32 \times 15 = 480 \]

### Step 3: Determine the effect of decreasing each observation by 7
If each observation is decreased by 7, the total decrease in the sum of observations will be:
\[ \text{Total decrease} = 7 \times \text{Number of observations} = 7 \times 15 = 105 \]

### Step 4: Calculate the new sum of observations
Now, we subtract the total decrease from the original sum of observations:
\[ \text{New sum of observations} = 480 - 105 = 375 \]

### Step 5: Calculate the new mean
To find the new mean, we use the new sum of observations and divide it by the number of observations:
\[ \text{New mean} = \frac{\text{New sum of observations}}{\text{Number of observations}} = \frac{375}{15} \]
Calculating this gives:
\[ \text{New mean} = 25 \]

### Final Answer
The resulting mean, if each observation is decreased by 7, is **25**.

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