Question

The mean of five observations is $5$ and their variance is $9.20.$ If three of the given five observations are $1, 3$ and $8$, then a ratio of other two observations is :

• A. 4:09
• B. 6:07
• C. 5:08
• D. 10:03

Answer 1

The Correct Option is A

The problem gives us the mean and variance of five observations, along with three specific observations. We need to find the ratio of the remaining two observations. Let's solve this step-by-step.

1. The mean of the five observations is given as 5.
   • If the observations are x_1, x_2, x_3, x_4, x_5, the mean is calculated as:
     \frac{x_1 + x_2 + x_3 + x_4 + x_5}{5} = 5
   • This implies:
     x_1 + x_2 + x_3 + x_4 + x_5 = 25
2. The variance of the observations is given as 9.20.
   • Variance is calculated using:
     \frac{\sum_{i=1}^{5} (x_i - 5)^2}{5} = 9.20
   • This implies:
     \sum_{i=1}^{5} (x_i - 5)^2 = 46
3. We know three of the observations: 1, 3, and 8.
   • Substituting these observations into the sum equation:
     1 + 3 + 8 + x_4 + x_5 = 25
   • This simplifies to:
     x_4 + x_5 = 13 (Equation 1)
4. Next, substitute these into the variance equation:
   • Calculate the deviations:
     (1-5)^2, (3-5)^2, (8-5)^2 which are 16, 4, and 9 respectively.
   • Add these deviations:
     16 + 4 + 9 + (x_4 - 5)^2 + (x_5 - 5)^2 = 46
   • Simplify the equation:
     29 + (x_4 - 5)^2 + (x_5 - 5)^2 = 46
   • Thus:
     (x_4 - 5)^2 + (x_5 - 5)^2 = 17 (Equation 2)
5. Now, consider solving Equations 1 and 2 to find the specific values of x_4 and x_5:
   • From Equation 1:
     x_4 = 13 - x_5
   • Substitute into Equation 2:
     \left((13 - x_5) - 5\right)^2 + (x_5 - 5)^2 = 17
   • Which simplifies to:
     \left(8 - x_5\right)^2 + (x_5 - 5)^2 = 17
   • Solving this quadratic equation will yield the values of x_4 and x_5. The calculated values (by further simplifying the equation) result in a viable ratio:
   • Finally, we find the ratio of x_4 to x_5: 4:9.

Therefore, the correct answer is 4:09.