Question

Can you answer the following showing formula and method used.

Diameters of bolts produced by a particular machine are normally distributed with mean 0.760cm and standard deviation 0.012cm. Specifications call for the diameters to range from 0.730cm to 0.780cm a) What proportion of bolts will be smaller than 0.740cm? b) What value of the diameter is exceeded by 10% of the bolts? c) What proportion of bolts will meet the specification? d) What value of the standard deviation is required so that less than 3% of the bolts have a diameter greater than 0.78cm?

Answer 1

a) The diameters of bolts follow a normal distribution with:
mean = 0.760
s = 0.012

a) P(X < 0.740) = P(z < (0.740-0.760)/0.012) = P(z < -1.67) = 0.0475

b) We need to find z0 when: P(z > z0) = 0.10
OR
P(z < z0) = 0.90
Hence, z0 = 1.29
(x - 0.760)/0.012 = 1.29
X = 0.776

c) P(0.730 < X < 0.780) = P( (0.730-0.760)/0.012 < z < (0.780 - 0.760)/0.012 ) = P(-2.5 < z < 1.67) = P(z < 1.67) - P(z < -2.5) = 0.9525 - 0.0062 = 0.9463

d) We need to find s when:
P(X > 0.78) < 0.03
P(z > (0.780-0.760)/s) < 0.03
P(z < 0.02/s) > 0.97
z0>=1.89
0.02/s >= 1.89
s = 0.011