Identifying Binomial Distributions. In Exercises 5–12, 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 least one requirement that is not satisfied.

Surveying Senators The Senate members of the 113th Congress include 80 males and 20 females. Forty different senators are randomly selected without replacement, and the gender of each selected senator is recorded.

Out of 80 senators, 40 are selected, and the gender of each selected senator is recorded.

The following assumptions of the binomial distribution should be satisfied:

• The procedure should have a fixed number of trials.
• The trials should be independent.
• Each trial should have outcomes that are of exactly two kinds: success and failure.
• The probability of success should be the same for all the trials.

One of the conditions that must be met for a procedure to follow the binomial distribution is that the events whose probability is computed should be independent.

Fortysenators are selected without replacement. Thus, they cannot be considered independent unless they fulfill the 5% rule of cumbersome calculations, which says that the sample size should be no more than 5% of the population size.

It is given that the population size is equal to 100.

The sample size chosen is equal to 40.

It is known that

$$\begin{gathered} 5\%\text{of}100=\frac{5}{100}\times 100 \\ =5 \end{gathered}$$

However,

$40>5$

Since the sample size is greater than 5% of the population size, the selections cannot be considered independent.

Therefore, the given situation cannot be modeled using the binomial distribution.