Identifying Binomial Distributions. In Exercises, determine whether the given procedure results in a binomial distribution (or a distribution that can be treated as binomial). For those that are not binomial, identify at hast one requirement that is not satisfied
Surveying Senators The current Senate consists of 83 males and 17 females. Forty different senators are randomly selected without replacement, and the gender of each selected senator is recorded.
To be a binomial probability distribution, a procedure must satisfy the preceding four requirements:
1. The procedure has a fixed number of trials. Here, 40 different senators were randomly selected.
2. The trials must be independent. Here, the population consists of 83 males and 17 females, that is, population size is 100. We are given that sample is selected without replacement. They are not independent. (Since the sample size is more than 5% of the population (40% of the population). From the 5% Guideline for Cumbersome Calculations, also they are not independent.)
3. Each trial must have all outcomes classified into two categories. Here, responses were recorded into two categories, male and female.
4. The probability of success remains the same in all trials.
Since trials are not independent, the second requirement is not satisfied. So, this is a distribution that cannot be treated as binomial.