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question about normal distribution?

There are two machines available for cutting corks intended for use in wine bottles. The first produces corks with diameters that are normally distributed with mean 3 cm and a standard deviation of 0.1 cm. The second machine produces corks with diameters that have a normal distribution with mean 3.05 cm and a standard deviation of 0.08 cm. Acceptable corks have diameters between 2.9 cm and 3.1 cm. Which machine is more likely to produce an acceptable cork?

we need P(2.9 < X < 3.1) ... if X is N(3 , 0.1)
and P(2.9 < Y < 3.1) ... if Y is N(3.05 , 0.08)

now, x1 = 2.9 , x2 = 3.1
z1 = (2.9 - 3) / 0.1 = -1
z2 = (3.1 - 3) / 0.1 = 1

P(-1 < Z < 1) = 2*0.8413 - 1 = 0.6826

now, y1 = 2.9 , y2 = 3.1
z3 = (2.9 - 3.05) / 0.08 = -1.875
z4 = (3.1 - 3.05) / 0.08 = 0.625

P(-1.875 < Z < 0.625) = 0.7357 - (1 - 0.9699) = 0.7056

Thus the second machine is more likely to produce such corks...

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