standard deviation find the probability that more than 30%

Question

A manufacturer produces a certain type of components that has a length of 2.1 cm. the distribution of this components is normal with mean 2.15 cm and sd 0.4cm. There are 50 pieces in a bag. Find the probability that a randomly selected bag will contain more than 30% of the components with length longer than 2.1cm. Thank You

Answer 1

Let L represent the length of a randomly chosen component, so that

 

L ~ N(μ=2.15, σ = 0.4).

 

A length of 2.1 centimeters has a standard score of

 

z = (L - μ)/σ = (2.1 - 2.15)/0.4 = -0.125

 

The probability of a single component being longer than 2.1cm is

 

P(L > 2.1) = P(z > -0.125) = 0.5497

 

So the number of components in the bag longer than 2.1cm is

 

X ~ Bin(n=50, p=0.5497)

 

There are more than 30% components in the bag of length L > 21cm with probability

 

P(X > 15) ≈ 0.9997