Question

A random sample of n people selected from a large population will be asked whether they have read a novel in the past year. Let the random variable R represent the number of people from the sample who answer yes. The variance of random variable R is 6. Assume the responses are independent of each other. If the proportion of people from the population who read a novel in the past year is 0.40, which of the following is the best interpretation of random variable R ?

1. A binomial variable with 15 independent trials
2. A binomial variable with 25 independent trials
3. A variable that is not binomial with 25 independent trials
4. A binomial variable with 40 independent trials
5. A variable that is not binomial with 40 independent trials

Answer 1

A binomial variable with 25 independent trials

Answer 2

Option 2) A binomial variable with 25 independent trials

Explanation:

We are given the following in the question:

Sample size = n

R: the number of people from the sample who answer yes to the question whether they have read a novel in the past year.

`var(R) = 6`

`p =0.40`

Then, the random variable R follows a binomial distribution:

• There are n independent trials.
• Each trial have two results either they have read the book or they have not read the book
• Probability of success for each trial is same.

`var(R) = npq\\6 = n(0.4)(1-0.4)\\\\n = (6)/(0.4* 0.6)\\\\n = 25`

Thus, R is a binomial variable with 25 independent trials.