The independent random variables L L\,L and S S\,S represent the masses, in kg, of large and small industrial components, such that
L∼N(6.5,0.82)andS∼N(2.1,0.52) L \sim \mathrm{N}(6.5, 0.8^2) \quad \text{and} \quad S \sim \mathrm{N}(2.1, 0.5^2) L∼N(6.5,0.82)andS∼N(2.1,0.52)The random variables L1,L2 L_1, L_2\,L1,L2 are independent and each has the same distribution as LLL. The random variables S1,S2,S3 S_1, S_2, S_3\,S1,S2,S3 are independent and each has the same distribution as SSS.
The random variable M M\,M is defined as the mean mass of a specific batch of these components:
M=L1+L2+S1+S2+S35 M = \frac{L_1 + L_2 + S_1 + S_2 + S_3}{5} M=5L1+L2+S1+S2+S3Find P(M>4.2)P(M > 4.2)P(M>4.2).
The random variable Q Q\,Q represents the mass of a quality-control weight, where Q∼N(μ,0.92)Q \sim \mathrm{N}(\mu, 0.9^2)Q∼N(μ,0.92).
Given that P(Q−L<−4.5)=0.22P(Q - L < -4.5) = 0.22P(Q−L<−4.5)=0.22 and that Q Q\,Q and L L\,L are independent,
find the value of μ\muμ, giving your answer to 3 significant figures.
