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Binomial assumptions For the following random variables, check whether the conditions needed to use the binomial distribution are satisfied or not. Explain a. \(X=\) number of people suffering from a contagious disease in a family of \(4,\) when the probability of catching this disease is \(2 \%\) in the whole population (binomial, \(n=4, p=0.02\) ). (Hint: Is the independence assumption plausible?) b. \(X=\) number of unmarried women in a sample of 100 females randomly selected from a large population where \(40 \%\) of women in the population are unmarried \((\) binomial, \(n=100, p=0.40)\) c. \(X=\) number of students who use an iPhone in a random sample of five students from a class of size \(20,\) when half the students use iPhones (binomial, \(n=5, p=0.50\) ). d. \(X=\) number of days in a week you go out for dinner. (Hint: Is the probability of dining out the same for each day?)

Step by step solution

01

Check Assumptions for Part a

For a binomial distribution, the trials must be independent, and each trial has the same probability of success. Here, each family member catching the disease could depend on whether another member is already infected, which might violate the independence assumption. However, if the probability of catching the disease from within the family is low because of strict isolation, we might assume independence.

02

Check Assumptions for Part b

In a sufficiently large population, the probability of selecting an unmarried woman should remain constant at 40%. The random selection of 100 females from this population satisfies the conditions of independence and constant probability of success, making the binomial distribution applicable.

03

Check Assumptions for Part c

The sample of five students is drawn from a class of 20, which is not a large population. The trials (picking students) might not be independent, particularly if the class composition of iPhone users is known. Since the sampling is without replacement from a small population, the condition for constant probability is likely violated here.

04

Check Assumptions for Part d

The assumption of a binomial distribution requires independent trials with a constant probability of success each time. Going out for dinner each day may depend on various factors (e.g., weekdays vs. weekends), potentially violating the assumption of constant probability. Thus, using a binomial distribution may not be suitable.

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Key Concepts

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

Random Variables

In probability and statistics, a random variable is a way to describe the outcomes of a random process. Consider it as a function that assigns a numerical value to every outcome in a sample space.
For example, imagine rolling a fair six-sided die. If we define a random variable, say \(X\), it could represent the number that appears on the die. This means \(X\) can take any integer value from 1 to 6, with each outcome having an equal chance of occurring.
Understanding random variables is crucial because they help us model real-world scenarios where randomness is involved. In our exercises, we examine specific random variables like:

• Number of people suffering from a disease in a family (Exercise a)
• Number of unmarried women in a sample (Exercise b)
• Number of iPhone users in a classroom sample (Exercise c)

All of these represent scenarios where the outcome is uncertain. Modeling these scenarios accurately with random variables allows for the application of probability distributions, like the binomial distribution, that help us understand and predict outcomes.

Probability of Success

The probability of success is central in using the binomial distribution. It refers to the likelihood of a specific outcome occurring in each trial of an experiment.
For a binomial distribution, every trial should have the same probability of success, indicated by symbol \(p\). This uniformity is necessary to apply binomial probability calculations effectively.
In our examples:

• In Exercise a, the probability of catching a disease is 2% or 0.02.
• Exercise b involves a 40% or 0.40 probability of selecting an unmarried woman.
• Exercise c deals with a 50% or 0.50 probability of choosing an iPhone user.

The idea is simple: Each trial needs to retain this probability to maintain consistency across all attempts in the experiment. If this criterion is met, calculating the probability of a specific number of successes becomes straightforward using the binomial formula.

Independence Assumption

The independence assumption is a fundamental aspect when using a binomial distribution. It requires that the outcome of one trial does not affect the outcome of another.
In other words, knowing the result of one trial should not give information about another trial's result.
Let's consider our scenarios:

• For Exercise a, catching a disease might not be independent if the disease transfers easily within a family. Independence could be assumed if contact within the family is minimal.
• Exercise b posits independence due to the random selection of each female from a large population, assuming no direct correlation.
• In Exercise c, selecting students without replacement from a small class makes independence tricky since each selection affects the probability of the next.

Maintaining independence ensures that the binomial model's predictions and calculations are valid and reliable. Deviations from this assumption can lead to inaccuracies in the analysis and conclusions.