Question

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 Ten different senators from the 113th Congress are randomly selected without replacement, and the numbers of terms that they have served are recorded.

Short answer

The given situation cannot be approximated using the binomial distribution as it consists of the following features which violate the assumptions of the binomial distribution:

• The number of terms that a senator has served has more than two possible outcomes.
• The sample size is greater than 5% of the population size, and, hence, the selections cannot be considered independent.

Step-by-step solution

Given information

A sample of 10 senators is selected randomly, and the number of terms they have served is recorded.

Assumptions of binomial distribution

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.

Violation of Assumptions of binomial distribution

First violation:

One of the conditions that must be met for a procedure to follow the binomial distribution is that the outcomes of the event whose probability is to be estimated must be of exactly two types.

One of the outcomes is regarded as a success, while the other is considered a failure.

Here, the number of terms that a senator has served has more than two possible outcomes.

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

Second violation:

Another key assumption is that the trials should be independent of each other.

Here, 10 senators 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 10.

It is known that

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

However,

$10>5$

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.