91Ó°ÊÓ

Simulation The exponential probability distribution can be used to model waiting time in line or the lifetime of electronic components. Its density function is skewed right. Suppose the wait time in a line can be modeled by the exponential distribution with \(\mu=\sigma=5\) minutes. (a) Simulate obtaining 100 simple random samples of size \(n=10\) from the population described. That is, simulate obtaining a simple random sample of 10 individuals waiting in a line where the wait time is expected to be 5 minutes. (b) Test the null hypothesis \(H_{0}: \mu=5\) versus the alternative \(H_{1}: \mu \neq 5\) for each of the 100 simulated simple random samples. (c) If we test this hypothesis at the \(\alpha=0.05\) level of significance, how many of the 100 samples would you expect to result in a Type I error? (d) Count the number of samples that lead to a rejection of the null hypothesis. Is it close to the expected value determined in part (c)? What might account for any discrepancies?

Step by step solution

01

Understand the Exponential Distribution

The exponential distribution models the time between events in a Poisson process. In this case, it models the wait time in a line, with the mean \(\mu\) and standard deviation \(\sigma\) both equal to 5 minutes.

02

Generate 100 Simple Random Samples

Use a statistical software or programming language (such as R, Python) to generate 100 random samples of size \( n=10 \) from the exponential distribution with \( \mu=\sigma=5 \). Each sample will represent the wait times of 10 individuals.

03

State the Hypotheses

The null hypothesis \( H_{0}: \mu=5 \) represents that the mean wait time is 5 minutes. The alternative hypothesis \( H_{1}: \mu \eq 5 \) represents that the mean wait time is not 5 minutes.

04

Perform Hypothesis Test for Each Sample

For each of the 100 samples, conduct a hypothesis test (e.g., a t-test) to test \( H_{0}: \mu=5 \) against \( H_{1}: \mu \eq 5 \) at the \( \alpha=0.05 \) level of significance. Record whether the null hypothesis is rejected or not.

05

Calculate Expected Type I Errors

A Type I error occurs when the null hypothesis is true, but we reject it. With \( \alpha=0.05 \), we expect 5% of the tests to result in a Type I error. For 100 samples, the expected number of Type I errors is \( 100 \times 0.05 = 5 \).

06

Count the Rejections

Count the number of times the null hypothesis was rejected in the 100 samples. This represents the number of samples that lead to a rejection of \( H_{0} \).

07

Compare Expected and Actual Rejections

Compare the actual number of rejections to the expected 5 rejections from Step 5. Discrepancies might arise due to random sample variability, but should generally be close if the samples are truly random and independent.

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.

Simulation

Simulation is a powerful technique used in statistics to mimic the behavior of various real-world processes. The main idea is to use a computational model to generate data samples that represent the actual scenario you are studying. For example, in the provided exercise, we simulated 100 simple random samples from an exponential distribution representing wait times.
Simulations can help visualize data, understand probabilistic behavior, and test hypotheses without needing to conduct expensive or time-consuming real-world experiments.

Use the following steps for the simulation in the exercise:

• Use a programming language like Python or R.
• Generate 100 sets of 10 random samples.
• Use the exponential distribution with \( \mu = 5 \).

This will help you see how wait times vary within small groups, providing a robust dataset to analyze.

Hypothesis Testing

Hypothesis testing is a statistical method that allows us to make inferences about population parameters based on sample data. In the context of the exercise, we test whether the mean wait time (\( \mu \)) is 5 minutes against the alternative that it is not.

Steps to perform hypothesis testing:

• State the null hypothesis \( H_0: \ \mu = 5 \) and the alternative hypothesis \(H_1: \ \mu \e 5 \).
• Select a significance level (\( \alpha = 0.05 \) in this exercise).
• Conduct a t-test for each random sample to check if you reject the null hypothesis.

By applying this procedure to each sample, you can determine whether the observed sample means differ significantly from the hypothesized mean of 5 minutes.

Type I Error

A Type I error occurs when the null hypothesis \( H_0 \) is incorrectly rejected when it is actually true. This is a false positive.
In our exercise, if we set \( \alpha = 0.05 \), the probability of committing a Type I error is 5% in each test.

Given 100 tests, we would expect about:
\[ 100 \times 0.05 = 5 \]
Type I errors. This means, out of 100 hypothesis tests, around 5 would typically falsely reject the true null hypothesis.

The main points to remember about Type I errors are:

• They represent a false alarm - detecting an effect that isn't there.
• The significance level \( \alpha \) controls the rate of Type I errors.
• In our example, setting \( \alpha = 0.05 \) implies we tolerate a 5% chance of this error.

Understanding Type I errors is crucial in hypothesis testing and in interpreting the results carefully.

Random Sample Generation

Generating random samples is a fundamental step in simulation and hypothesis testing. Random sampling ensures each member of the population has an equal chance of being included, leading to unbiased and representative samples.

To generate random samples from an exponential distribution with \( \mu = 5 \) minutes, follow these steps:

• Use a programming language like Python or R, which have built-in functions for this purpose.
• In Python, for instance, you can use the \texttt{numpy.random.exponential(scale, size)} function where scale = 5 and size = (100, 10) for 100 samples of size 10.
• Ensure proper random seed initialization for reproducibility by using \texttt{numpy.random.seed()}.

By following these steps, you generate the necessary random samples to proceed with your analysis. Random sampling is a cornerstone in statistics to avoid bias and achieve reliable results.