Question

In Exercises \(\mathrm{P} .95\) to \(\mathrm{P} .99,\) determine whether the process describes a binomial random variable. If it is binomial, give values for \(n\) and \(p .\) If it is not binomial, state why not. Worldwide, the proportion of babies who are boys is about 0.51 . We randomly sample 100 babies born and count the number of boys.

Short answer

This process describes a binomial random variable. The number of trials \(n\) is 100 and the probability of success \(p\) is 0.51.

Step-by-step solution

Determine the fixed number of trials

In the given scenario, 100 babies are randomly sampled. So, the number of trials is fixed and defined as \(n=100\).

Determine the success and failure

The scenario deals with determining the gender of babies which has only two possible outcomes - either the baby is a boy (success) or not (failure). Hence, the second condition of a binomial process is also satisfied.

Verify the independence of trials

Since the births of individuals are independent events, selecting one baby does not affect the probability of the gender of the next baby. Thus, the trials are independent.

Check uniform probability of success

The problem states that the global proportion of male babies is approximately 0.51. This probability remains the same for every birth, satisfying the fourth condition of a binomial process. Hence, the probability of success is defined as \(p=0.51\).

Final evaluation

Since all four conditions for a binomial process are met, we can conclude that the described scenario represents a binomial random variable.