Determine whether each random variable described below satisfies the conditions for a binomial setting

Determine whether each random variable described below satisfies the conditions for a binomial setting, a geometric setting, or neither. Support your conclusion in each case.

(a) A high school principal goes to 10 different classrooms and randomly selects one student from each class. X – the number of female students in his group of 10 students.
(b) You are on Interstate 80 in Pennsylvania, counting the occupants in every fifth car you pass. Let Z – the number of cars you pass before you see one with more than two occupants.

The correct answer and explanation is :

Let’s analyze each scenario to determine if it fits a binomial setting, a geometric setting, or neither, based on the following criteria:

Binomial Setting Criteria:

1. Fixed number of trials, n.
2. Each trial has two possible outcomes: “success” or “failure”.
3. Trials are independent.
4. Probability of success is the same for each trial.

Geometric Setting Criteria:

1. Trials are independent.
2. Each trial has two outcomes: “success” or “failure”.
3. Probability of success is constant.
4. The variable of interest is the number of trials until the first success.

(a) X – the number of female students in a group of 10 selected students from different classrooms

This scenario fits a binomial setting.

• There is a fixed number of trials: 10 students are chosen.
• Each trial results in a success (female student) or failure (not female).
• Selections are made independently by choosing one student from different classes, which likely ensures minimal dependence.
• The probability of selecting a female student is assumed to be constant across classrooms.

Therefore, X is the number of successes (female students) in a fixed number of independent, identical trials — the key definition of a binomial distribution.

(b) Z – the number of cars passed before seeing one with more than two occupants

This scenario fits a geometric setting.

• Each car can be a success (more than two occupants) or failure (two or fewer occupants).
• You’re observing each car independently as you pass them.
• The probability of success is assumed to be the same for each car, since you’re sampling systematically (every fifth car).
• The variable Z is defined as the number of cars passed until the first success.

These are precisely the conditions that define a geometric setting.

Conclusion:

• (a) Binomial setting.
• (b) Geometric setting.

Both settings align with their respective statistical models based on the nature of the trials and what is being measured.