According to New Jersey Transit, the 8=00 a. M. weekday...

According to New Jersey Transit, the $8=00$ a. $\mathrm{M}$. weekday train from Princeton to New York City has a $90 \%$ chance of arriving on time To test this claims, an auditor chooses 6 weekdays at random during a month to ride this train. The train arrives late an 2 of those days. Does the auditor have convincing evidence that the company's claim isn't true? Design and carry out a simulation to answer this question. Follow the four-step process.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about a population parameter based on sample data. In this context, a null hypothesis is assumed (such as the claim that the train arrives on time 90% of the time) and an alternative hypothesis is proposed (that the actual on-time arrival percentage is less than 90%). The testing process involves comparing observed data to what would be expected under the null hypothesis to determine whether the observed discrepancies are statistically significant.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has only two outcomes: success or failure. Here, 'success' could be defined as the train arriving on time. Since the auditor takes a fixed number of trips and each trip is an independent event with a known probability of success, the binomial distribution is the appropriate model to determine the likelihood of seeing a certain number of on-time arrivals.

Simulation (Monte Carlo Simulation)

Simulation, particularly Monte Carlo simulation, involves using computer-generated random samples to mimic the behavior of a real-world process or experiment. In the scenario of testing the train's on-time arrival claim, simulation allows us to model the distribution of outcomes (e.g., the number of on-time arrivals) under the null hypothesis assumption, thereby providing an empirical method to estimate probabilities and assess the credibility of the company's claim.

P-value

The p-value is a statistical measure that quantifies the probability of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is true. In this case, it represents the likelihood of observing two or fewer on-time arrivals in a sample of six, if the true probability of an on-time arrival is 90%. A very low p-value would indicate that such an outcome is highly unlikely under the null hypothesis, thereby providing evidence against the company’s claim.

Four-Step Process for Simulation

The four-step process for simulation typically involves: (1) defining the problem and formulating the hypotheses, (2) planning the simulation including specifying the probability model and simulation procedure, (3) executing the simulation to generate a distribution of outcomes under the null hypothesis, and (4) analyzing the results to draw conclusions about the hypothesis being tested. This systematic approach ensures that the simulation study is rigorous and that the conclusions drawn are based on well-structured statistical evidence.

Transcript

00:02 According to the new jersey transit, a specific train has a 90 % chance of arriving on time.

00:10 However, one specific person observed that six weekdays, when he fixed six weekdays at random, the train arrived late two of those days.

00:22 So train was late two out of six days.

00:29 So we want to test whether or not this claim that the train is on time, 90 % of the time, is actually true.

00:38 So how do we do this? let's design and carry out the simulations to be following the four -step process.

00:44 So what are the four steps? well, first we have state.

00:48 So state the question.

00:50 So we want to test, but we want to know what is the chance that the train, is late two out of six days if it is on time 90 % of the time.

01:14 So we assume it's on time 90 % of the time and what's the probability that we observe this specific case that's late two out of six days.

01:25 So let's plan.

01:27 Plan our simulation.

01:28 How do we simulate this? well, what we could do is we could generate a random number, or let's generate six random numbers, actually, one for each day, six random digits, so in between zero and nine.

01:53 So, and then we say, if the digit is anything but zero, the train was on time.

02:09 So this is a valid simulation for this 90 % chance of arriving on time because you can check, well, the probability of a digit being zero is one out of the total number of digits, which is 10.

02:27 So the probability that it's not on time is 0 .1.

02:33 Okay, so we generate our six random digits in each one...

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