91Ó°ÊÓ

For each situation, identify the sample size \(n\), the probability of a success \(p\), and the number of success \(x\). When asked for the probability, state the answer in the form \(b(n, p, x)\). There is no need to give the numerical value of the probability. Assume the conditions for a binomial experiment are satisfied. Since the Surgeon General's Report on Smoking and Health in 1964 linked smoking to adverse health effects, the rate of smoking the United States have been falling. According to the Centers for Disease Control and Prevention in 2016, \(15 \%\) of U.S. adults smoked cigarettes (down from \(42 \%\) in the \(1960 \mathrm{~s}\) ). a. If 30 Americans are randomly selected, what is the probability that exactly 10 are smokers? b. If 30 Americans are randomly selected, what is the probability that exactly 25 are not smokers?

Step by step solution

01

Solution for part a

In this scenario, the sample size \(n\) is the number of Americans randomly selected, which is 30. The probability of a success \(p\) (being a smoker in this case) is given as \(15\%\), which can be represented as \(0.15\) in decimal form. The number of successes \(x\) is the number of smokers we are interested in, which is 10. From these numbers, the answer should be written in the form \(b(n, p, x)\), hence the probability asked for here is represented as \(b(30, 0.15, 10)\).

02

Solution for part b

In this scenario, \(n\) is still the number of Americans randomly selected, which is 30. However, this time \(p\) (being a non-smoker in this case) is not directly given. We should consider that the sum of the probabilities of all possible outcomes must be \(1\). Since we know that the probability of being a smoker is \(0.15\), the probability of being a non-smoker would be \(1 - 0.15 = 0.85\). The number of successes \(x\) in this scenario is the number of non-smokers we are interested in, which is 25. Hence the probability asked for here is represented as \(b(30, 0.85, 25)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Size

The sample size, often denoted as , is a fundamental component in statistics that underlines how many observations or trials are performed in an experiment. When considering a binomial probability scenario, such as determining the likelihood that a certain number of individuals will exhibit a particular behavior, it's essential to keep in mind that the sample size should be fixed in advance.

For instance, in the given exercise where we're looking at the probability of Americans being smokers or non-smokers, the sample size is 30. This means that the calculations for the probability will be based on exactly 30 people. The choice of sample size can heavily influence the precision and reliability of the study's results. Larger sample sizes generally provide more reliable estimates and reduce the margin of error in probability outcomes.

Probability of Success

When dealing with binomial experiments, the 'probability of success,' symbolized as p, is the chance that a particular outcome we deem as 'success' will occur in a single trial. 'Success' here doesn't necessarily mean a positive result; it simply refers to the outcome of interest.

In the exercise, success is defined as an individual being a smoker, with a probability of success (p) of 15%, or 0.15 in decimal form. It's critical that this probability remains constant for each trial. Understanding how different probabilities of success affect the overall outcome is crucial, as it allows one to comprehend the dynamics between rare and common events in the probability distribution.

Number of Successes

The number of successes in a binomial experiment, denoted by x, refers to how many times the specific event we're tracking occurs. It's the count we're interested in when it comes to the outcomes of our trials.

For example, in part a of our exercise, the number of successes is the scenario where exactly 10 out of the 30 individuals selected are smokers. It is crucial to clearly define what is considered a 'success' in your experiment, so that the number of successes can be accurately counted. This count of successes is what we use to calculate the binomial probability and analyze the data.

Binomial Experiment Conditions

Binomial experiments have specific conditions that must be met in order to apply the binomial probability formula effectively. These conditions include:
- The number of observations or trials, n, is fixed.
- Each trial can result in just two possible outcomes: success or failure.
- The probability of success, p, remains the same for each trial.
- The trials are independent; the outcome of one trial does not affect the others.

In our exercise, the conditions for a binomial experiment are satisfied, as we have a fixed number of trials (30 people), two outcomes (smoker or non-smoker), a constant probability of being a smoker or non-smoker, and the assumption that each individual's smoking status is independent of anyone else's. This framework is pivotal for estimating the probability of exactly how many successes will occur over a series of trials.