Question

2. When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is 5%. Let ? = the number of defective boards in a random sample of size ? = 25.

   a. Determine ? (? = 2) (Using the formula of the pmf) b. Determine ? (1 ≤ ? ≤ 4) (Using the table). c. Calculate the expected value d. Calculate variance and standard deviation of ?.

Answer 1

Answer:

Given that,

When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is 5%.

Let ? = the number of defective boards in a random sample of size ? = 25.

Given that circuit boards used in the manufacture of compact disc players are tested and the long-run percentage of defectives is 5%.

Probability that a circuit board is defective is=p=5%=0.05

Now a random sample of n=25 defective boards are to be tested.

X=Number of defective boards in a random sample of size n=25

Before we go on to solve the problem let us know a bit about Binomial Distribution

Binomial Distribution:

A discrete random variable X is said to have a binomial distribution if its probability mass function(PMF) is given by,

Where 0<p<1

Notation :

Expected value or Mean:

E(X)=np

Variance and Standard Deviation:

Variance(X)=np(1-p)

Standard deviation(X)=

Coming back to our problem:

X= Number of defective boards in a random sample of size n=25

Probability that a circuit board is defective is=p=5%=0.05

(a).

Here we need to determine P(X=2):

(b).

Here we need to determine,

P(1 X 4) =P(X = 1)+ P(X = 2) +P(X = 3)+ P(X = 4)

(c).

Here we need to calculate the expected value.

We know that,

E(X)=n p=25 0.05=1.25

Hence the expected value of X is 1.25

(d).

Here we need to calculate the variance and standard deviation of X.

We know that,

Hence,

Variance(X)=1.1875 and

Standard Deviation(X)=1.0897