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Chapter 8: Problem 8

Sampling Gravestones. The local genealogical society in Coles County, Illinois, has compiled records on all 55,914 gravestones in cemeteries in the county for the years 1825 through 1985 . Historians plan to use these records to learn about African Americans in Coles County's history. They first choose an SRS of 395 records to check their accuracy by visiting the actual gravestones. 3 a. How would you label the 55,914 records? b. Use software, the Simple Random Sample applet, or Table \(\underline{B}\) (starting at line 141 ) to choose the first six records for the SRS.

Short Answer

Expert verified
Label records 00001 to 55914; select six random samples using Table B, starting at line 141.

Step by step solution

01

Label the Records

To label the 55,914 gravestones for the purpose of creating a simple random sample, assign each gravestone a unique integer label. These labels should range from 00001 to 55914. The use of leading zeroes ensures that each label has the same number of digits, which helps avoid any bias in the selection process.
02

Understand the Sampling Method

When choosing a Simple Random Sample (SRS), each sample of a given size has distinct integers that represent the gravestones. The sample candidates are then randomly chosen such that every possible sample of that size has an equal chance of being selected.
03

Use Table B Starting at Line 141

Table B is a table of random digits used in simple random sampling. To begin, locate the starting point at line 141. Read sequential sections, skipping any that don't correspond to numbers between 00001 and 55914, until you have chosen six unique numbers.
04

Select the First Six Records

Proceeding from line 141, reference the numbers provided in Table B. Select numbers with five digit representations ensuring they fall within the interval of 00001 to 55914. Repeat this process to identify six distinct numbers which will represent the records in your SRS.

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Key Concepts

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

Labeling Records
To start the process of creating a simple random sample, you need to systematically label every record you're studying. For the gravestones in Coles County, each of the 55,914 records receives a unique identifier. This is done by assigning a number ranging from 00001 to 55914. Using leading zeroes ensures each number has the same number of digits, which helps to organize the data uniformly. This uniform numeric label makes the selection process unbiased and easier to manage. By starting your labels uniformly from 00001, you ensure all records are equally represented, paving the way for an unbiased sampling process.
Random Digits Table
A Random Digits Table is a handy tool when selecting samples without bias. It is essentially a long list of numbers, each digit chosen randomly, which means no patterns influence your choice. In the given exercise, you would use Table B, starting from line 141, to select numbers that will represent the records. By reading the numbers line by line and ignoring those that do not lie between 00001 and 55914, you efficiently find random records. Make sure to choose numbers that fit within this range to maintain the randomness and relevance of the sample. The process mimics modern computational random number generation methods in a more manual format.
Sampling Methods
Sampling methods help researchers select elements from a larger population to better understand the whole. In this exercise, a Simple Random Sample (SRS) is used. SRS is a basic sampling method where every possible sample of a given size has an equal chance of being selected. Here, each of the 395 records chosen to check for accuracy is selected without bias or influence, relying purely on chance. This method is valuable because it provides a representative picture without systematic bias, leading to more credible and reliable results when analyzing historical data.
Sample Size
Deciding the sample size is crucial in statistical analysis. In this situation, historians choose a sample size of 395 records from the entire 55,914. This number was likely determined based on logistical feasibility and the necessity to strike a balance between manageability and reliability of data. The sample size should be large enough to give a good representation of the whole set of records but not so large that it becomes impractical to handle during the field verification process. A well-chosen sample size like 395 helps ensure accurate, actionable insights without undue expenditure of time and resources.
Bias Avoidance
Avoiding bias is the cornerstone of credible research. The primary goal in using a method like simple random sampling is to ensure each record has an equal opportunity to be selected. This opposes any external factors that might skew the results. For instance, without labeling the records in a uniform manner or using a truly random selection process (like a random digits table), some records might be favored over others, leading to skewed data. Proper labeling and a meticulous selection process enable historians to avoid biases that could distort findings about African American history in Coles County, ensuring the integrity of their research outcomes.

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Most popular questions from this chapter

The Pew Research Center Report titled "Libraries 2016," released September 9,2016 , asked a random sample of 1601 Americans aged 16 and over, "Have you personally ever visited a public library or used a public library bookmobile in person in the last 12 months?" In the entire sample, \(48 \%\) said Yes. But only \(40 \%\) of those in the sample over 65 years of age said Yes. Which of these two sample percentages will be more accurate as an estimate of the truth about the population? a. The result for those over 65 is more accurate because it is easier to estimate a proportion for a small group of people. b. The result for the entire sample is more accurate because it comes from a larger sample. c. Both are equally accurate because both come from the same sample.

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?

Student Opinions. A university has 30,000 undergraduate and 10,000 graduate students. A survey of student opinion concerning health care benefits for domestic partners of students selects 300 of the 30,000 undergraduate students at random and then separately selects 100 of the 10,000 graduate students at random. The 400 students chosen make up the sample. a. What is the probability that any of the 30,000 undergraduates is in your random sample of 300 undergraduates selected? What is the probability that any of the 10,000 graduate students is in your random sample of 100 graduate students selected? b. If you have done the calculations correctly in part (a), the probability of any student at the university being selected is the same. Why is your sample of 400 students from the university not an SRS of students? Explain.

Air port Shuttle. Blue Ribbon taxis offers shuttle service to the nearest airport. You look up the online reviews for Blue Ribbon taxis and find that there are 17 reviews, 6 of which report that the taxi never showed up. Is this a biased sampling method for obtaining customer opinion on the taxi service? If so, what is the likely direction of bias? Explain your reasoning carefully.

More on Random Digit Dialing. By mid-2017, about \(53.9 \%\) of adults lived in households with a cell phone and no landline phone. Among adults aged 25-29, this percentage was about \(73.3 \%\), while among adults over 65 , the percentage was only \(23.9 \% .\) a. Write a survey question for which the opinions of adults with landline phones only are likely to differ from the opinions of adults with cell phones only. Give the direction of the difference of opinion. b. For the survey question in part (a), suppose a survey were conducted using random digit dialing of landline phones only. Would the results be biased? What would be the direction of bias? c. Most surveys now supplement the landline sample contacted by RDD with a second sample of respondents reached through random dialing of cell phone numbers. The landline respondents are weighted to take account of household size and number of telephone lines into the residence, whereas the cell phone respondents are weighted according to whether they were reachable only by cell phone or also by landline. Explain why it is important to include both a landline sample and a cell phone sample. Why is the number of telephone lines into the residence important? (Hint: How does the number of telephone lines into the residence affect the chance of the household being included in the RDD sample?)
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