The mean and variance of 5 observations are 6 and 6.8 respectively. If a number equal to mean is included in the set of observations is k, then the value of `(34)/(k)` is equal to

Answer

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

### Step 1: Understand the Given Information
We are given:
- The mean of 5 observations is \( \bar{x} = 6 \).
- The variance of these observations is \( \sigma^2 = 6.8 \).

### Step 2: Calculate the Sum of the Observations
The mean is defined as the sum of the observations divided by the number of observations. Therefore, we can express the sum of the observations \( S \) as follows:
\[ S = 5 \times \bar{x} = 5 \times 6 = 30 \]

### Step 3: Calculate the Sum of the Squares of the Observations
The variance is defined as the average of the squares of the observations minus the square of the mean. We can express this as:
\[ \sigma^2 = \frac{S^2}{n} - \bar{x}^2 \]
Rearranging this gives:
\[ S^2 = n \cdot \sigma^2 + n \cdot \bar{x}^2 \]
Substituting the values we have:
\[ S^2 = 5 \cdot 6.8 + 5 \cdot 6^2 \]
Calculating \( S^2 \):
\[ S^2 = 5 \cdot 6.8 + 5 \cdot 36 = 34 + 180 = 214 \]

### Step 4: Calculate the Sum of the Squares of the Observations
Now we know:
\[ S^2 = s_1^2 + s_2^2 + s_3^2 + s_4^2 + s_5^2 = 214 \]

### Step 5: Include the New Observation
Now, we include a new observation \( k \) which is equal to the mean (6). The new sum of observations becomes:
\[ S' = S + k = 30 + 6 = 36 \]
The new mean \( \bar{x}' \) is:
\[ \bar{x}' = \frac{S'}{6} = \frac{36}{6} = 6 \]

### Step 6: Calculate the New Variance
The new variance \( \sigma'^2 \) can be calculated as:
\[ \sigma'^2 = \frac{s_1^2 + s_2^2 + s_3^2 + s_4^2 + s_5^2 + k^2}{6} - (\bar{x}')^2 \]
Substituting the known values:
\[ \sigma'^2 = \frac{214 + 6^2}{6} - 6^2 \]
Calculating:
\[ \sigma'^2 = \frac{214 + 36}{6} - 36 = \frac{250}{6} - 36 = \frac{250 - 216}{6} = \frac{34}{6} \approx 5.67 \]

### Step 7: Find the Value of \( \frac{34}{k} \)
Since \( k = 6 \):
\[ \frac{34}{k} = \frac{34}{6} = \frac{17}{3} \approx 5.67 \]

### Final Answer
Thus, the value of \( \frac{34}{k} \) is \( \frac{17}{3} \).

---