A square inch of carpeting contains 50 carpet fibers. The

Answer

Step 1 of 2

(a)

A square inch of carpeting contains  carpet fibers.

The probability of a damaged fiber is  Assume that the damaged fibers occur independently.

We are asked to approximate probability of one or more damaged fibers in one square yard of carpeting.

We need to use the normal approximation to the binomial.

Number of fibers used in the

Let  denote the number of damaged fibers in

Hence  is binomial with

We need to find the probability

We will use approximation here which can be defined as,

If  is a binomial random variable with parameters  and ,

Is approximately a standard normal random variable.

To approximate a binomial probability with a normal distribution, a continuity correction is applied as follows:

The approximation is good for

Since  we can use the approximation.

 Hence the probability  we can write,

We can write  as,

From the z table, the area to the left of  is .

Hence the approximate probability of one or more damaged fibers in one square yard of carpeting is