SOLUTION TO PROBLEM 27 OF CHAPTER 4
The total number, X, of defective bolts in a shipment of 10,000 has the binomial
distribution with parameters n= 10000 and p=0.05. We want to find the smallest
integer aso that P(X>a)≤0.01. We will apply Chebychev’s Inequality, (26) on
page 101, to estimate a.
We ha ve µ=np = 500 and σ2=np(1 −p) = 475. Thus
P(X>a)≤P(X≥a)=P(X−500 ≥a−500)
≤P(|X−500|≥a−500) ≤475
(a−500)2.
Solving the inequality 475/(a−500)2≤0.01 for the smallest integer a, we find that
a= 718. Note: In Exercise 46 of Chapter 7, we will use the Central Limit Theorem
to get a significantly better estimate of a.