The mean of 40 observations 20 and their standard deviation is 5. If the sum of the square of the observations k, then the value of `(k)/(1000)` is

Answer

To solve the problem step by step, we will use the information given about the mean, standard deviation, and the number of observations.

### Step 1: Understand the given information
We are given:
- Number of observations (n) = 40
- Mean (x̄) = 20
- Standard deviation (σ) = 5

### Step 2: Write the formula for standard deviation
The formula for standard deviation is:
\[ \sigma = \sqrt{\frac{\sum (x_i^2)}{n} - \left(\frac{\sum x_i}{n}\right)^2} \]
Where:
- \( \sum (x_i^2) \) is the sum of the squares of the observations.
- \( \sum x_i \) is the sum of the observations.

### Step 3: Calculate the sum of observations
Since the mean is given, we can find the sum of the observations:
\[ \sum x_i = n \cdot x̄ = 40 \cdot 20 = 800 \]

### Step 4: Substitute values into the standard deviation formula
Now, substituting the known values into the standard deviation formula:
\[ 5 = \sqrt{\frac{\sum (x_i^2)}{40} - \left(\frac{800}{40}\right)^2} \]
Calculating \( \left(\frac{800}{40}\right)^2 \):
\[ \left(\frac{800}{40}\right)^2 = 20^2 = 400 \]
Thus, the equation becomes:
\[ 5 = \sqrt{\frac{\sum (x_i^2)}{40} - 400} \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides gives:
\[ 25 = \frac{\sum (x_i^2)}{40} - 400 \]

### Step 6: Rearrange the equation to solve for \( \sum (x_i^2) \)
Adding 400 to both sides:
\[ 25 + 400 = \frac{\sum (x_i^2)}{40} \]
\[ 425 = \frac{\sum (x_i^2)}{40} \]

### Step 7: Multiply both sides by 40
To find \( \sum (x_i^2) \):
\[ \sum (x_i^2) = 425 \cdot 40 = 17000 \]

### Step 8: Identify \( k \)
From the problem, we know that \( k = \sum (x_i^2) = 17000 \).

### Step 9: Calculate \( \frac{k}{1000} \)
Finally, we need to find:
\[ \frac{k}{1000} = \frac{17000}{1000} = 17 \]

### Final Answer
Thus, the value of \( \frac{k}{1000} \) is **17**.

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