Exams Joint Entrance Examination Mathematics Statistics Question 72 Question 72 of 208 Hard MCQ
Let the mean and standard deviation of marks of class A of 100 students be respectively 40 and α ( > 0 ) \alpha(>0) α ( > 0 ) , and the mean and standard deviation of marks of class B of n students be respectively 55 and 30 − α 30-\alpha 30 − α . If the mean and variance of the marks of the combined class of 100 + n 100+n 100 + n students are respectively 50 and 350, then the sum of variances of classes A and B is:
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\[ A B A + B x ‾ 1 = 40 x ‾ 2 = 55 x ‾ = 50 σ 1 = α σ 2 = 30 − α σ 2 = 350 n 1 = 100 n 2 = n 100 + n x ‾ = 100 × 40 + 55 n 100 + n \begin{array}{|cccccccccc|}\quad \mathbf{A} & \mathbf{B} & \mathbf{A}+\mathbf{B} \\ \overline{\mathrm{x}}_{1}=40 & \overline{\mathrm{x}}_{2}=55 & \overline{\mathrm{x}}=50 \\ \sigma_{1}=\alpha & \sigma_{2}=30-\alpha & \sigma^{2}=350 \\ \mathrm{n}_{1}=100 & \mathrm{n}_{2}=\mathrm{n} & 100+\mathrm{n} \\ \\ \overline{\mathrm{x}}=\frac{100 \times 40+55 \mathrm{n}}{100+\mathrm{n}} &\end{array} A x 1 = 40 σ 1 = α n 1 = 100 x = 100 + n 100 × 40 + 55 n B x 2 = 55 σ 2 = 30 − α n 2 = n A + B x = 50 σ 2 = 350 100 + n \]
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5000 + 50 n = 4000 + 55 n 1000 = 5 n n = 200 σ 1 2 = ∑ x i 2 100 − 40 2 σ 2 2 = ∑ x j 2 100 − 55 2 350 = σ 2 = ∑ x i 2 + ∑ x j 2 300 − ( x ‾ ) 2 350 = ( 1600 + α 2 ) × 100 + [ ( 30 − α ) 2 + 3025 ] × 200 300 − ( 50 ) 2 2850 × 3 = α 2 + 2 ( 30 − α ) 2 + 1600 + 6050 8550 = α 2 + 2 ( 30 − α ) 2 + 7650 α 2 + 2 ( 30 − α ) 2 = 900 α 2 − 40 α + 300 = 0 α = 10 , 30 σ 1 2 + σ 2 2 = 10 2 + 20 2 = 500 \begin{aligned}
& 5000+50 \mathrm{n}=4000+55 \mathrm{n} \\\\
& 1000=5 \mathrm{n} \\\\
& \mathrm{n}=200 \\\\
& \sigma_{1}^{2}=\frac{\sum \mathrm{x}_{\mathrm{i}}^{2}}{100}-40^{2} \\\\
& \sigma_{2}^{2}=\frac{\sum \mathrm{x}_{\mathrm{j}}^{2}}{100}-55^{2} \\\\
& 350=\sigma^{2}=\frac{\sum \mathrm{x}_{\mathrm{i}}{ }^{2}+\sum \mathrm{x}_{\mathrm{j}}^{2}}{300}-(\overline{\mathrm{x}})^{2} \\\\
& 350=\frac{\left(1600+\alpha^{2}\right) \times 100+\left[(30-\alpha)^{2}+3025\right] \times 200}{300}-(50)^{2} \\\\
& 2850 \times 3=\alpha^{2}+2(30-\alpha)^{2}+1600+6050 \\\\
& 8550=\alpha^{2}+2(30-\alpha)^{2}+7650 \\\\
& \alpha^{2}+2(30-\alpha)^{2}=900 \\\\
& \alpha^{2}-40 \alpha+300=0 \\\\
& \alpha=10,30 \\\\
& \sigma_{1}^{2}+\sigma_{2}^{2}=10^{2}+20^{2}=500
\end{aligned} 5000 + 50 n = 4000 + 55 n 1000 = 5 n n = 200 σ 1 2 = 100 ∑ x i 2 − 4 0 2 σ 2 2 = 100 ∑ x j 2 − 5 5 2 350 = σ 2 = 300 ∑ x i 2 + ∑ x j 2 − ( x ) 2 350 = 300 ( 1600 + α 2 ) × 100 + [ ( 30 − α ) 2 + 3025 ] × 200 − ( 50 ) 2 2850 × 3 = α 2 + 2 ( 30 − α ) 2 + 1600 + 6050 8550 = α 2 + 2 ( 30 − α ) 2 + 7650 α 2 + 2 ( 30 − α ) 2 = 900 α 2 − 40 α + 300 = 0 α = 10 , 30 σ 1 2 + σ 2 2 = 1 0 2 + 2 0 2 = 500
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