1. Manufacture of a certain component requires three different machining operati

Question

1. Manufacture of a certain component requires three different machining operations. Suppose that the amount of time for each operation is normally distributed, and that the three macn ing times are independent of one another. Further suppose that the mean machining times for the three operations are 15, 30, and 20 minutes, respectively, with standard deviations of 1, 2, and 1.5 minutes, respectively. a) What is the distribution of the total machining time needed to produce a single compo nent? (b) What is the probability that it takes at most one hour to produce a cornponent? (cSuppose there is an order for seven of these components to be produced. What is the distribution of the total time needed to produce all seven components? In an 8-hour shift, the machinists get a twenty-minute lunch break and two more ten-minute breaks; what is the probability that the order can be completed by the end of the shift?

Explanation / Answer

a) let X1 , X2 and X3 be the time taken to complete three operation

X1 - N(15,1^2)

X2 - N(30,2^2)

X1 - N(20,1.5^2)

Y = X1+X2+X3

Y - N(15 +30 +20 , 1^2 + 2^2 + 1.5^2)

= N(65,7.25)

b)

P(Y < 60)

P(Z< ( 60 - 65)/sqrt(7.25))

= P(Z< -1.85695)

= 0.0317

c)

S = 7 Y

S - N(7*65 , 7^2*7.25)

= N(455,355.25)

total time = 8*60 - 20 -2*10 = 440

P(S < 440)

=P(Z< ( 440 -455)/sqrt(355.25))

=P(Z< -0.79583)

= 0.2131

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