A particular program say program A previously drew 30 of the

A particular program, say, program A, previously drew 30% of the television audience. A random sample of 100 viewers who currently watch program A in the 100 viewers. Is X a binomial random variable? Briefly explain your answer. If yes, identify the n and P. What is the normal distribution by which X is approximated? (Be specific about the parameters.) Be sure to ensure that the necessary conditions for this approximation are satisfied. What is the probability that fewer than 105 out of a sample of 100 will be found to watch the program? Be sure to include a correction for continuity.

Solution

Q6.
a.
Yes, It is binomial. We only have 2 possibble outcomes

Binomial Experiment-
A binomial experiment (also known as a Bernoulli trial) is a statistical experiment that has the following properties:
1. The experiment consists of n repeated trials.
2. Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
3. The probability of success, denoted by P, is the same on every trial.
4. The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

n=100, p = 0.30

b.

n*p>5, 100*0.3> 5 => 30>5
n*(1-p)>5, 100*0.7> 5 => 30>5
Can Use Normal Approximation                  
                  
c.
Normal Approximation to Binomial Distribution
Mean ( np ) =400 * 0.3 = 120
Standard Deviation ( npq )= 400*0.3*0.7 = 9.1652
Normal Distribution = Z= X- u / sd                   
P(X < 105) = (105-120)/9.1652
= -15/9.1652= -1.6366
= P ( Z <-1.6366) From Standard NOrmal Table
= 0.0509