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Chapter 11: Problem 13

Roadblock State police set up a roadblock to estimate the percentage of cars with up-to-date registration, insurance, and safety inspection stickers. It would be too inconvenicnt and costly to check every vehicle that passes through a checkpoint, so they decide to stop about \(1 / 20\) of the vehicles. a) Why would a simple random sample be unreasonable for this situation. b) Identify two possible sampling schemes that could be used. Explain how randomization would be used in each.

Short Answer

Expert verified
a) Simple random sampling is impractical due to logistic and safety issues. b) Systematic sampling (stop every 20th car) or cluster sampling (check all cars in a lane or time slot).

Step by step solution

01

Understanding Simple Random Sampling

In a simple random sample, each vehicle has an equal chance of being chosen, against which every other car stopped is unlikely to be practical at a roadblock. Pulling over cars randomly from a moving traffic flow without introducing additional traffic issues or ensuring driver safety would be problematic.
02

Problem with Simple Random Sampling

Implementing simple random sampling in a dynamic setting like moving traffic can create logistical challenges, traffic snarls, safety issues, and could lead to non-compliance, making it an impractical choice.
03

Suggesting a Systematic Sampling Scheme

One possible approach is systematic sampling. For example, officers could stop every 20th car. By starting from a randomly chosen vehicle near the beginning of the roadblock, they create a uniform interval system ensuring that approximately 1/20 of vehicles is checked.
04

Explaining Systematic Sampling

Officers initiate counting from a randomly chosen starting point, selecting every 20th vehicle thereafter. This method evenly distributes the sampling across the stream of cars with reduced operational complexity compared to completely random stops.
05

Suggesting a Cluster Sampling Scheme

Another technique is cluster sampling. Randomly choosing a time period or lane (cluster) during which all cars passing through are stopped and checked. For example, all vehicles in the right lane for 15 minutes.
06

Explaining Cluster Sampling

In this scheme, officers select clusters of cars determined by random choice of time or traffic lane, thus randomizing the sample by changing the time or lane regularly, allowing for practicality and efficient coverage with minimal traffic disruption.

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Key Concepts

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

Systematic Sampling
Systematic sampling is a practical way to select samples from a larger group in an orderly manner, especially in situations where random sampling is cumbersome. In the context of a roadblock, systematic sampling involves selecting every nth vehicle for inspection. For instance, stopping every 20th car. This approach ensures that a uniform portion of the traffic is sampled.

Here's how it works:
  • Select a random starting point by picking a number between 1 and 20. Suppose the number 7 is chosen, start checking from the 7th car.
  • From that starting point, select every 20th vehicle for inspection.
  • This method spreads out the sampled cars over the entire time span, improving logistical handling.
Systematic sampling significantly reduces the operational complexity compared to simple random sampling. It avoids traffic congestion caused by abrupt stopping and starting, which is important for maintaining safety and traffic flow at the checkpoint.
Cluster Sampling
Cluster sampling is another effective method especially suitable in situations like traffic checkpoints where samples can be naturally grouped. For a roadblock, this means dividing the traffic into clusters such as specific lanes or time periods. Once a cluster is randomly selected, all vehicles within that cluster are checked.

The procedure could look like this:
  • Identify clusters such as different lanes or predetermined time slots.
  • Randomly select a cluster, e.g., conduct checks on all cars in the left lane for the next 15 minutes.
  • Inspect every vehicle passing through the selected cluster.
Cluster sampling effectively leverages natural groupings within the flow of traffic, reducing the logistical burden of stopping vehicles randomly. The method allows for the collection of comprehensive data from each selected cluster while minimizing disruption to normal traffic flow.
Simple Random Sampling
Simple random sampling involves selecting samples purely at random, where each subject from the population has an equal chance of being chosen. At a roadblock, however, implementing such a method is impractical.

Here are the issues associated with simple random sampling in this context:
  • Difficulty in managing random stops without causing traffic disruption.
  • Problems ensuring safety and compliance, as random stopping can surprise drivers, potentially leading to accidents.
  • Logistical challenges due to the fast-paced nature of moving traffic, making it difficult to coordinate random checks effectively.
While simple random sampling is ideal for studies where conditions allow for truly random selections without impacts on the environment or population being sampled, it proves to be less viable in traffic or road checkpoints due to the inherent chaos such an approach might introduce.

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Most popular questions from this chapter

\- Parent opinion, part 1 In a large city school system with 20 elementary schools, the school board is considering the adoption of a new policy that would require elementary students to pass a test in order to be promoted to the next grade. The PTA wants to find out whether parents agree with this plan. Listed below are some of the ideas proposed for gathering data. For each, indicate what kind of sampling strategy is involved and what (if any) biases might result. a) Put a big ad in the newspaper asking people to log their opinions on the PTA website. b) Randomly select one of the elementary schools and contact every parent by phone. c) Send a survey home with every student, and ask parents to fill it out and return it the next day. d) Randomly select 20 parents from each elementary school. Send them a survey, and follow up with a phone call if they do not return the survey within a week.

Satisfied workers The managers of a large company wished to know the percentage of employees who feel "extremely satisfied" to work there. The company has roughly 24,000 employees. They contacted a random sample of employees and asked them about their job satisfaction, obtaining 437 completed responses. a) The company's annual report states, "Our survey shows that \(87.34 \%\) of our employees are "very happy" working here." Comment on that claim. Use appropriate statistics terminology. b) One manager suggested surveying employees by assigning computer-generated random numbers to each employee on a list of all employees and then contacting all those whose assigned random number is divisible by \(7 .\) Is this a simple random sample? c) For each scenario suggested by a different manager, determine the sampling method. i. Use the company e-mail directory to contact 150 cmployees from among those employed for less than 5 years, 150 from among those employed for \(5-10\) years, and 150 from among those employed for more than 10 years. ii. Use the company e-mail directory to contact every 50th employce on the list. iii. Select several divisions of the company at random. Within each division, draw an SRS of employees to contact. d) One manager suggested having the head of each corporate division hold a meeting of their employees to ask whether they are happy on their jobs. They will ask people to raise their hands to indicate whether they are happy. What problems do you see with this plan? e) For each of these designs proposed by a different manager, identify the problem with the method and the effect it would have on the estimate of the percentage of cmployees who feel "extremely satisfied" to work there. i. Leave a stack of surveys out in the employee cafeteria so people can pick them up and return them. ii. Stuff a questionnaire in the mailbox of each employee with the request that they fill it out and return it.

Drug tests Major League Baseball tests players to see whether they are using performance-enhancing drugs. Officials select a team at random, and a drug- testing crew shows up unannounced to test all 40 players on the team. Each testing day can be considered a study of drug use in Major League Baseball. a) What kind of sample is this? b) Is that choice appropriate?

Roper Through their Roper Reports Worldwide, GfK Roper conducts a global consumer survey to help multinational companies understand different consumer attitudes throughout the world. Within 30 countries, the researchers interview 1000 people aged \(13-65 .\) Their samples are designed so that they get 500 males and 500 females in each country. (www.gfkamerica.com) a) Are they using a simple random sample? Explain. b) What kind of design do you think they are using?

\- Sampling methods Consider each of these situations. Do you think the proposed sampling method is appropriate? Explain. a) We want to know what percentage of local doctors accept Medicaid patients. We call the offices of 50 doctors randomly selected from local Yellow Page listings. b) We want to know what percentage of local businesses anticipate hiring additional employees in the upcoming month. We randomly select a page in the Yellow Pages and call every business listed there.
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