An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2. The variance of marks obtained by 30 girls is also 2. The average marks of all 50 candidates is 15. If μ is the average marks of girls and σ2 is the variance of marks of 50 candidates, then μ + σ2 is equal to _________.
σ2b= 2 (variance of boys), n1 = no. of boys
\(\bar{b_g}\) = 12, n2 = no. of girls
σ2g = 2
\(\bar{x_g}\) = \(\frac{50 \times 15 - 12 \times σ_b}{30}\)
= \(\frac{750 - 12 \times 20}{30}\) = 17 = μ
variance of combined series
σ2 = \(\frac{n_1σ_b^2}{n_2σ^2_g}\) + \(\frac{n_1.n_2}{(n_1 + n_2)^2}(\bar{_b} - \bar{x_g}^2)\)
σ2 = \(\frac{20 \times 2 + 30 \times 2}{20+30}\) + \(\frac{20 \times 30}{(20 + 30)^2}(12 - 17)^2\)
σ2 = 8
⇒ μ + σ2 = 17 + 8 = 25