Question

The average diameters of the thickness of the diameters of a large number of screws are normally distributed with an average equal to 2.4 cm and standard deviation equal to 0.5 cm
a) What fraction of screws will have an average diameter greater than 3 cm?
b) If the screws that have an average diameter equal to or less than 1.9 cm are discarded, what percentage is eliminated?
c) It is assumed that three screws are randomly selected from all, what is the probability that all three have an average diameter greater than 3 cm?
d) Below what measure will be the diameters of the screws that have the smallest 15% of the diameters?

Answer 1

Let X denotes the diameter of a randomly selected screw.

X ~ Normal(2.4, 0.5²)

a) The fraction of screws will have an average diameter greater than 3 cm

b)

The percentage of screws that is eliminated = 100*0.1587 = 15.87% 16%

c)

Let Y denotes the number of screws that have an average diameter greater than 3 cm out of randomly selected three screws.

Y ~ Binomial(n = 3, p = 0.1151)

The probability mass function of Y is

Now,

The probability that all three have an average diameter greater than 3 cm

d)