In Problems 7–16, determine which of the following probability...

Question

In Problems 7–16, determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why.

One hundred randomly selected U.S. parents with at least one child under the age of 18 are surveyed and asked if they have ever spanked their child. The number of parents who have spanked their child is recorded.

Accepted Answer

Welcome back everyone. In this problem, a researcher randomly selects 50 households in a large city and records whether each household owns at least 1 electric vehicle. Is this a binomial experiment? Select the best answer. A says no, this is not a binomial experiment because the trials are not independent. B says, yes, this is a binomial experiment because all the conditions are satisfied. C says no, this is not a binomial experiment because the number of trials is not fixed, and the D says yes, this is a binomial experiment because there are only two possible outcomes. Now, how do we know if this scenario represents a binomial experiment? Well, let's first ask ourselves what do we know about these types of experiments. Well, we know that a binomial experiment has to have a fixed number of trials. OK. We know that it must have two possible outcomes. That's why it's named binomial, OK. We know that there has to be a constant probability of success. And we know that there has to be independence in the experiment. So the question we now have to ask ourselves is from the problem statement, does it satisfy all of these conditions? Well, for starters, we're told that the researcher randomly selects 50 households. So that means that the number of trials is fixed, OK, because he's using 50 households or trials. Next, we're told that. The researchers regarding whether each household owns at least one electric vehicle, and the answer to that question is either yes or no. Either you own at least one electric vehicle, that's a success, or you do not. That is the failure. So it has two possible outcomes. Next, is there a constant probability of success? Well, yes, the probability of a household owning at least one electric vehicle is constant across all trials or researchers select these households at random. Uh, if one household has an electric vehicle, at least one electric vehicle, it has no bearing and if the other household will also have one. And finally, That is why this child satisfies independence. The outcome for each household is assumed to be independent of others. Therefore, since all the conditions for a binomial experiment are met, this scenario represents a binomial experiment. Thus, B is the correct answer. We can be sure we are right by reviewing the remaining choices. A says it isn't a binomial experiment because the trials are not independent, but they are, assuming one whole's electric vehicle ownership doesn't affect others. C says it's not a binomial experiment because the number of trials is not fixed, but we've already seen that it is fixed at 50, and D says it is a binomial experiment because there are only two possible outcomes. Now, while there are two possible outcomes, the experiment must also meet the other criteria. In other words, meeting only this criteria would not make it a binomial experiment. It has to have fixed trials, constant probability, and independence, which are all met in this case. So that's why D would not be the best answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Binomial Experiment Criteria

Independence of Trials

Fixed Number of Trials and Constant Probability

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