30. In an assembly-line production of industrial robots, gearbox assemblies can

Question

30. In an assembly-line production of industrial robots, gearbox assemblies can be installed in 1 minute each if holes have been properly drilled in the boxes and in 10 minutes each if the holes must be redrilled. Twenty gearboxes are in stock, and it is assumed that two will have improperly drilled holes. Five gearboxes are to be randomly from the 20 available for installation in the next five robots in line. (a) Find the probability that all five boxes will fit properly (b) Find the expected value, variance, and the standard deviation of the time it takes to install these five gearboxes.

Explanation / Answer

Solution:

Let X = # improperly drilled gearboxes.

Then, X HG(N = 20, r = 2 (as success for nonconforming boxes, n = 5)

with PMF p(x) =([ 2Cx ] [ 18C5-x ]) / [ 20Cx ] for x = 0,1,2.

(a) P(all gearboxes fit properly) = P(X = 0) = p(0) = ([2C0 ])[18C 5 ])/ ( 20C5 ) = 0.5526.

Hence, P(all gearboxes fit properly) = 0.5526.

(b) Since X denotes the number of improperly drilled gearboxes, hence

time needed to install the gearboxes = T = [(5 X) * 1 ]+ X * 10 = 9X+5 minutes.

Now, E(X) = (n * r) /N = (5 * 2)/ 20 = 0.5

and V (X) = n *( r/ N) *[(Nr)/ n] *[ (Nn)/( N1)] = 5 *( 2/ 20) *( 18/ 20) *( 15/ 19) = 0.3553.

Therefore, E(T) = E(5 + 9X) = 5 + 9E(X) = 5 + 9 * 0.5 = 9.5,

and V (T) = V (5 + 9X) = 92 *V (X) = 81 * 0.3553 = 28.7793

and SD(X) = sqrt( V (X)) = 5.3646

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