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: Identify the given values
- Number of observations (n) = 40
- Mean (x̄) = 20
- Standard deviation (σ) = 5

### Step 2: Use the formula for standard deviation
The formula for the standard deviation in terms of the sum of squares of observations is given by:

\[ \sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2 \]

Where:
- \(\sigma^2\) is the variance (which is the square of the standard deviation)
- \(\sum x_i^2\) is the sum of the squares of the observations
- \(n\) is the number of observations
- \(\bar{x}\) is the mean

### Step 3: Substitute the known values into the formula
We know that:
- \(\sigma = 5\), so \(\sigma^2 = 5^2 = 25\)
- \(\bar{x} = 20\), so \(\bar{x}^2 = 20^2 = 400\)

Substituting these values into the formula gives:

\[ 25 = \frac{\sum x_i^2}{40} - 400 \]

### Step 4: Rearrange the equation to find \(\sum x_i^2\)
Rearranging the equation:

\[ 25 + 400 = \frac{\sum x_i^2}{40} \]

\[ 425 = \frac{\sum x_i^2}{40} \]

### Step 5: Multiply both sides by 40
To isolate \(\sum x_i^2\):

\[ \sum x_i^2 = 425 \times 40 \]

### Step 6: Calculate the sum of squares
Calculating the right side:

\[ \sum x_i^2 = 17000 \]

### Step 7: Relate the sum of squares to k
According to the problem, the sum of the squares of the observations is denoted as \(k\). Therefore:

\[ k = 17000 \]

### Step 8: Find \(\frac{k}{1000}\)
Now we need to find \(\frac{k}{1000}\):

\[ \frac{k}{1000} = \frac{17000}{1000} = 17 \]

### Final Answer
The value of \(\frac{k}{1000}\) is \(17\).

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