Benzene (C6H6 – the simplest aromatic hydrocarbon), commonly used to synthesize

Question

Benzene (C6H6 – the simplest aromatic hydrocarbon), commonly used to synthesize plastics in industry, was discovered by Michael Faraday in 1825 in illuminating gas made from whale oil. The hydrogentation of benzene to cyclohexane is promoted with a finely divided porous nickel catalyst. The catalyst particles can be considered to be spheres of various sizes. Assume all the particles have integer masses between 10 and 30 g. Let X be the mass of a randomly chosen particle. The probability mass function of X is given by the formula shown below. What proportion of particles has masses less than 17 g? Also, provide the mean and standard deviation of the particle masses.

p(X) = (x – 10) / 190 for x = 11, 12, 13, …, 29 0 other wise

Explanation / Answer

Using P(X) = (x-10)/190, consider the table:

Thus,  

P(x<17) = P(11) + P(12) +P(13) + P(14) +P(15) + P(16)= 0.110526316 [ANSWER]

********************************************
  
E(x) = Expected value = mean = Sum(xP(x)) =    23 [ANSWER]

Var(x) = E(x^2) - E(x)^2 =    21

s(x) = sqrt [Var(x)] =    4.582575695 [ANSWER, STANDARD DEVIATION]

x P(x) x P(x) x^2 P(x) 11 0.005263 0.057895 0.636842 12 0.010526 0.126316 1.515789 13 0.015789 0.205263 2.668421 14 0.021053 0.294737 4.126316 15 0.026316 0.394737 5.921053 16 0.031579 0.505263 8.084211 17 0.036842 0.626316 10.64737 18 0.042105 0.757895 13.64211 19 0.047368 0.9 17.1 20 0.052632 1.052632 21.05263 21 0.057895 1.215789 25.53158 22 0.063158 1.389474 30.56842 23 0.068421 1.573684 36.19474 24 0.073684 1.768421 42.44211 25 0.078947 1.973684 49.34211 26 0.084211 2.189474 56.92632 27 0.089474 2.415789 65.22632 28 0.094737 2.652632 74.27368 29 0.1 2.9 84.1 Totals 23 550 =E(x) =E(x^2)

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