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Chapter 9: Problem 44

Suppose the arrival of cars at Burger King's drive-through follows a Poisson process with \(\mu=4\) cars every 10 minutes. (a) Simulate obtaining 30 samples of size \(n=40\) from this population. (b) Construct \(90 \%\) confidence intervals for each of the 30 samples. [Note: \(\sigma=\sqrt{\mu}\) in a Poisson process.] (c) How many of the intervals do you expect to include the population mean? How many actually contain the population mean?

Short Answer

Expert verified
Simulate 30 samples, construct confidence intervals, and count how many contain the mean. Expect 27 intervals to include the mean, and compare with the actual count.

Step by step solution

01

- Understand the Poisson Process

In a Poisson process with rate \(\mu = 4\) cars every 10 minutes, the number of cars arriving in any given amount of time follows a Poisson distribution with mean \(\mu\).
02

- Simulate 30 Samples

Use statistical software or a programming language to simulate obtaining 30 samples of size \(n = 40\) from a Poisson distribution with \(\text{mean} = 4\).
03

- Calculate Sample Means and Standard Errors

For each sample, calculate the sample mean \(\bar{x}\) and the standard error for the mean. Since the standard deviation \(\sigma\) of a Poisson process is \(\tra_bold =\ \sqrt{\mu}\), the standard error is given by \(\frac{\sqrt{\mu}}{\frasallis{r}{nr}}\).
04

- Construct 90% Confidence Intervals for Each Sample

Use the sample means and standard errors to construct 90% confidence intervals. A 90% confidence interval for a sample mean \(\bar{x}\) is given by \(\bar x \inol{\bar}s \bar{1\)}\ Neighborhood lawcalculator law 1 . 647\right).'
05

- Count Intervals Containing the Population Mean

Compare each of the 30 confidence intervals with the population mean (4). Count how many of the intervals include the population mean.
06

- Determine Expected and Actual Counts

Determine the number of confidence intervals you expect to include the population mean at a 90% confidence level, which is \(\text{90\}\text{30 = 27\). Then, compare it with the actual count from the previous step.

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Key Concepts

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

Poisson process
The Poisson process is a key concept in probability and statistics. It describes events happening independently over a period of time or space. The rate at which these events occur is denoted by \(\text{\mu}\). In our exercise, \(\text{\mu} = 4\) cars arrive at the drive-through every 10 minutes. This means that over any given 10-minute period, the expected number of arriving cars follows a Poisson distribution with a mean of 4.

Key characteristics of a Poisson process include:
  • Events are independent: The occurrence of one event does not affect the probability of another.
  • Events occur at a constant rate: The rate \(\text{\mu}\) is constant over time.
  • Only one event can occur at a time.
In this exercise, understanding the Poisson process helps us know how to simulate car arrivals and calculate relevant statistics.
confidence intervals
Confidence intervals allow us to estimate a population parameter based on sample data. In our problem, we use 90% confidence intervals to estimate the mean number of cars arriving at the drive-through (which is \(\text{\mu} = 4\)).

To construct a confidence interval:
  • Calculate the sample mean \(\bar{x}\) for each sample.
  • Determine the standard error (\(\text{s.e.}\)), which is \[ \text{s.e.} = \frac{\sqrt{\mu}}{\sqrt{n}} \] for this Poisson process.
  • Use the formula \(\bar{x} \pm z \cdot \text{s.e.}\), where \(z\) is the z-score corresponding to the desired confidence level (1.645 for 90%).
This gives a range where we expect the true population mean to lie. With a 90% confidence level, we expect that 90% of the intervals will contain the true mean of \(\text{\mu} = 4\).
sample simulation
To simulate samples from a population, we use computational tools or statistical software. In our problem, we simulate 30 samples, each of size \(n = 40\), from a Poisson distribution with a mean of 4.

The reasons for simulating samples include:
  • Understanding variability: Shows the variation we might expect in real sampling.
  • Practice statistical methods: Simulations help us practice constructing confidence intervals and other statistical estimations.
By simulating these samples, we gather data to understand the behavior of the Poisson distribution and apply statistical methods like calculating means and constructing confidence intervals.
standard error
The standard error is a crucial concept in statistics. It measures how much a sample mean is expected to vary from the true population mean. In our exercise with a Poisson process, the standard error is calculated using the formula:
\[ \text{Standard Error} = \frac{\sqrt{\mu}}{\sqrt{n}} \]
In this problem, \(\sqrt{\mu} = \sqrt{4} = 2\) and \(\sqrt{n} = \sqrt{40}\). Therefore, the standard error is \(\frac{2}{\sqrt{40}}\).
  • Sample mean variability: The smaller the standard error, the less the sample mean varies from the true mean.
  • Role in confidence intervals: It's used to calculate the range of our confidence interval for the sample mean.
Understanding the standard error helps us grasp how much the sample statistics can be expected to deviate from the actual population parameter.

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Most popular questions from this chapter

Construct the appropriate confidence interval. A simple random sample of size \(n=40\) is drawn from a population. The sample mean is found to be \(\bar{x}=120.5\) and the sample standard deviation is found to be \(s=12.9 .\) Construct a \(99 \%\) confidence interval about the population mean.

Affirmative Action A sociologist wishes to conduct a poll to estimate the percentage of Americans who favor affirmative action programs for women and minorities for admission to colleges and universities. What sample size should be obtained if she wishes the estimate to be within 4 percentage points with \(90 \%\) confidence if (a) she uses a 2003 estimate of \(55 \%\) obtained from a Gallup Youth Survey? (b) she does not use any prior estimates?

A simple random sample of size \(n\) is drawn from a population that is normally distributed with population standard deviation, \(\sigma,\) known to be \(13 .\) The sample mean, \(\bar{x},\) is found to be 108. (a) Compute the \(96 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is 25 (b) Compute the \(96 \%\) confidence interval about \(\mu\) if the sample size, \(n\), is \(10 .\) How does decreasing the sample size affect the margin of error, \(E ?\) (c) Compute the \(88 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is \(25 .\) Compare the results to those obtained in part (a). How does decreasing the level of confidence affect the size of the margin of error, \(E ?\) (d) Could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed? Why? (e) Suppose an analysis of the sample data revealed three outliers greater than the mean. How would this affect the confidence interval?

A student constructs a \(95 \%\) confidence interval for the mean age of students at his college. The lower bound is 21.4 years and the upper bound is 28.8 years. He interprets the interval as, "There is a \(95 \%\) probability that the mean age of a student is between 21.4 years and 28.8 years" What is wrong with this interpretation? What could be done to increase the precision of the interval?

Severe acute respiratory syndrome (or SARS) is a viral respiratory illness. It has the distinction of being the first new communicable disease of the 21st century. Researchers wanted to estimate the incubation period of patients with SARS. Based on interviews with 81 SARS patients, they found that the mean incubation period was 4.6 days with a standard deviation of 15.9 days. Based on this information, construct a \(95 \%\) confidence interval for the mean incubation period of the SARS virus. Interpret the interval. (Source: Gabriel M. Leung et al., The Epidemiology of Severe Acute Respiratory Syndrome in the 2003 Hong Kong Epidemic: An Analysis of All 1755 Patients, Annals of Internal Medicine, \(2004 ; 141: 662-673 .)\)
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