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Chapter 6: Problem 72

determine whether the given random variable has a binomial distribution. Justify your answer Taking the train According to New Jersey Transit, the 8: 00 A.M. weekday train from Princeton to New York City has a \(90 \%\) chance of arriving on time on a randomly selected day. Suppose this claim is true. Choose 6 days at random. Let \(W=\) the number of days on which the train arrives late.

Short Answer

Expert verified
Yes, \(W\) has a binomial distribution.

Step by step solution

01

Define the Random Variable

The random variable in this context is defined as \(W\), which represents the number of days out of 6 that the train is late.
02

Identify the Criteria for a Binomial Distribution

A random variable has a binomial distribution if it satisfies four conditions: (1) There are a fixed number of trials, \(n\). (2) Each trial has only two possible outcomes: success or failure. (3) The probability of success, \(p\), is the same for each trial. (4) The trials are independent.
03

Verify the Fixed Number of Trials

In the problem, the number of days chosen is 6. Therefore, there are a fixed number of trials, \(n = 6\).
04

Determine the Two Outcomes

The two possible outcomes for each trial (day) are: (1) the train arrives on time (success), or (2) the train is late (failure).
05

Check the Consistent Probability of Success

According to the problem, the train has a 90% chance of arriving on time, hence a consistent probability of success, \(p = 0.9\), across all trials.
06

Assess the Independence of Trials

The probability of the train arriving on time (or being late) on one day does not affect the probability on another day, assuming independence between the days.
07

Conclusion

The random variable \(W\) meets all criteria for a binomial distribution: a fixed number of trials (6 days), two outcomes (on time or late), the same probability of success each time (90%), and independent trials. Therefore, \(W\) can be described by a binomial distribution.

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Key Concepts

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

Random Variables
In statistics, a random variable is a value determined by a set of possible outcomes from an experiment. It assigns a numerical value to each outcome, helping quantify random processes. In our exercise, the random variable is defined as \( W \), which counts the number of days the train arrives late out of 6 chosen days. This factors into understanding the distribution of outcomes in any chosen experiment.
  • A random variable can be discrete, having specific possible values like integers.
  • Alternatively, it can be continuous, where it has an infinite range of possibilities within a range.
The task defines \( W \) as discrete, allowing us to evaluate each specific day's outcome (late or on time).
Probability
Probability measures how likely an event is to occur. It ranges from 0 to 1, where 0 means the event will not happen and 1 means it certainly will. In the given problem, the probability, denoted as \( p \), that the train arrives on time is 90% or 0.9. Hence, the probability of the train being late is \( 1 - 0.9 = 0.1 \).Understanding probability helps determine the likelihood of different outcomes in our binomial distribution. It simplifies predicting the behavior of \( W \), allowing us to calculate the chances of the train arriving late on any given day out of the 6 observed.
  • The more consistent a probability, the more predictable and useful the model.
  • Probabilities help compare real-world outcomes with expected ones, verifying hypotheses.
Statistical Independence
Statistical independence implies that the outcome of one event does not influence another. Here, the arrival of the train one day doesn't affect its timeliness the next day. Independence is crucial to a binomial distribution because it means each trial outcome must be unaffected by others. By assuming independence, we simplify the model by ensuring the chance (90% on time) remains constant without external effects. Checks on whether independence holds true help validate the assumptions of the model.
  • If trials are dependent, separate models or adjustments may be needed.
  • Ensuring independence boosts the accuracy of predictions in similar statistical trials.
Discrete Probability Distribution
A discrete probability distribution outlines possible values of a random variable and their probabilities. It reflects scenarios where outcomes are finite or countable. Our exercise assumes such a distribution, where we're interested in counts of specific events (train's late days). For binomial distributions, we focus on repeated success-failure trials.The probability of \( W \) can be represented as a series of successes (train on time) or failures (train late) across the trials.
  • Discrete distributions map out likelihoods for each specific number of successes.
  • They allow us to calculate important statistical metrics like expected value and variance.
In practice, they help us understand the behavior of random variables under specific conditions.

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