The independent random variables MAM_AMA and MBM_BMB represent the masses, in grams, of two chemical reagents used in a laboratory process:
MA∼N(80,82)andMB∼N(35,42) M_A \sim \text{N}(80, 8^2) \quad \text{and} \quad M_B \sim \text{N}(35, 4^2) MA∼N(80,82)andMB∼N(35,42)A residual mass RRR is calculated using the formula R=2MA−4MBR = 2M_A - 4M_BR=2MA−4MB.
Determine the probability that the residual mass is less than 15 grams, P(R<15)P(R < 15)P(R<15).
A third reagent, a catalyst CCC, has a mass distributed as C∼N(50,σ2)C \sim \text{N}(50, \sigma^2)C∼N(50,σ2). Three independent samples of the catalyst, C1,C2, and C3C_1, C_2, \text{ and } C_3C1,C2, and C3, are selected and combined with one sample each of reagents MAM_AMA and MBM_BMB.
The random variable TTT is defined as the total mass of the combined mixture: T=MA+MB+∑i=13CiT = M_A + M_B + \sum_{i=1}^3 C_iT=MA+MB+∑i=13Ci.
Given that the probability of the total mass exceeding 290 grams is 0.0228, and assuming all reagent masses are independent,
calculate the value of σ\sigmaσ, the standard deviation of the catalyst mass.
