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Chapter 8: Problem 22

You must choose an SRS of 10 of the 440 retail outlets in New York that sell your company's products. How would you label this population to select a simple random sample? a. \(001,002,003, \ldots, 439,440\) b. \(000,001,002, \ldots, 439,440\) c. \(1,2, \ldots, 439,440\)

Short Answer

Expert verified
Label using 001, 002, ..., 440.

Step by step solution

01

Understand the Population Size

First, recognize that there are 440 retail outlets. This means that each retail outlet needs a unique numerical label for identification to be included in the random selection.
02

Identify the Necessity of Zero Padding

Since the total number is 440, the labels should accommodate three digits, as the maximum number of digits needed to express any label (in this case, '440') is three digits. This standardizes the length of the numeric label, facilitating a fair random selection process.
03

Analyze the Labeling Options

Review the given labeling options: - Option a: Labels use a three-digit format starting from 001 to 440. - Option b: Labels use a three-digit format starting from 000 to 440. - Option c: Labels do not use a consistent digit length (starts from 1). Option a maximizes consistency without unnecessary digits (000 is not required as there's no zeroeth retail outlet).
04

Conclusion on Correct Labeling

Since option a provides the most streamlined and straightforward numbering system without extra digits or inconsistencies (no outlet would be zero), it is the best choice for labeling. Labeling should start at 001 and go up to 440.

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Key Concepts

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

Population Labeling
In sampling techniques, particularly in simple random sampling (SRS), effective population labeling is crucial. Correct labeling involves assigning a unique identifier to each element of the population, which in this case are the retail outlets. By labeling each item, it ensures that every member of the population has an equal chance of being selected during the sampling process.

There are a few key factors to consider when labeling a population:
  • Each identifier should be unique to avoid any confusion during the sampling process.
  • Consistency in labeling ensures that comparison and selection procedures are easier and error-free.
  • The labels should accurately reflect the population size and characteristics.
By using a structured and consistent labeling method, such as using numbers from 001 to 440, it avoids unnecessary complications and makes the random sampling process straightforward and fair.
Zero Padding
Zero padding is a simple yet effective method to maintain consistency in label lengths. When dealing with numbers, especially in a sequence, zero padding involves adding leading zeros to numbers that do not have the maximum required digits.

For example, in labeling the retail outlets, using a three-digit format like 001, 002,..., 440 ensures that all numbers have the same number of digits. This consistency facilitates easier sorting and comparison:
  • Maintains uniformity across all labels allowing computer systems to automatically process and sort data efficiently.
  • Minimizes human error when manually handling data, as each label will have the same length.
  • Enhances readability, ensuring that the system used for selection does not encounter issues with varying label lengths.
Overall, zero padding is vital when numerical labels need to maintain a uniform appearance, especially when the numbers involved vary widely in their length.
Sampling Techniques
Sampling techniques are methods used to select individual units from a larger population. Simple Random Sampling (SRS) is one of the most straightforward techniques where every member of the population has an equal chance of selection.

SRS is favored for its simplicity and fairness, ensuring unbiased representation of the population. Here's how it works:
  • Every member of the population is listed with a unique identifier, like the labels we discussed earlier.
  • From these labeled identifiers, samples are selected randomly, ensuring no favoritism or bias occurs.
  • The random selection can be done using tools like a random number generator or drawing lots.
By using SRS, you ensure that the sample accurately reflects the broader population, which is essential for various analyses and decision-making processes. It's often used when the population is homogenous and lists or databases of individuals are easily accessible.

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Most popular questions from this chapter

Running Red Lights. A survey about driving habits by academic researchers produced a list of 5024 licensed drivers. 28 The investigators chose an SRS of 880 of these drivers to answer questions about their driving habits. a. How would you assign labels to the 5024 drivers? Choose the first 5 drivers in the sample. If you use Table B, start at line 118 . b. One question asked was, "Recalling the last ten traffic lights you drove through, how many of them were red when you entered the intersections?" Of the 880 respondents, 171 admitted that at least one light had been red. A practical problem with this survey is that people may not give truthful answers. What is the likely direction of the bias: Do you think more or fewer than 171 of the 880 respondents really ran a red light? Why?

Archaeologists plan to examine a sample of 2-metersquare plots near an ancient Greek city for artifacts visible in the ground. They choose separate samples of plots from floodplain, coast, foothills, and high hills. What kind of sample is this? a. A simple random sample b. A stratified random sample c. A voluntary response sample

Seat Belt Use. A study in El Paso, Texas, looked at seat belt use by drivers. Drivers were observed at randomly chosen convenience stores. After they left their cars, they were invited to answer questions that included questions about seat belt use. In all, \(75 \%\) said they always used seat belts, yet only \(61.5 \%\) were wearing seat belts when they pulled into the store parking lots.29 Explain the reason for the bias observed in responses to the survey. Do you expect bias in the same direction in most surveys about seat belt use?

Cluster Sampling. Cluster sampling begins by dividing the population into separate groups, or clusters. An SRS of the clusters is selected, and individuals in the cluster are sampled. If all the individuals in a cluster are sampled, this is called one-stage cluster sampling. If a random sample of individuals in a cluster is sampled, this is called two-stage sampling. Cluster sampling can be convenient when the individuals in a cluster are easily sampled as a group, such as all people in a neighborhood for a door-to- door survey. Here is a simple example of one-stage cluster sampling. All students at a small college are required to live in dormitories. There are 25 such dormitories on campus, each with 30 students. a. To select a cluster sample of 150 students, do the following. Label the dormitories from 01 to 25. Choose an SRS of 5 dormitories from the list of the 25. If you use Table \(\mathrm{B}\), enter the table at line 121 and indicate which dormitories you selected. Your cluster sample is the 150 students in these dormitories. b. How many dormitories would you have to sample if you wanted a sample of 100 students?

An opinion poll calls 2000 randomly chosen residential telephone numbers in Portland and asks to speak with an adult member of the household. The interviewer asks, "How many movies have you watched in a movie theater in the past 12 months?" In all, 831 people respond. The sample in this study is a. all adults living in Portland. b. the 2000 residential phone numbers called. c. the 831 people who responded.
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