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Chapter 1: Problem 65

A petition with 500 signatures is submitted to a college's student council. The council president would like to determine what proportion of those who signed the petition are actually registered students at the college. There is not enough time to check all 500 names with the registrar, so the council president decides to select a simple random sample of 30 signatures. Describe how she might do this.

Short Answer

Expert verified
The president should list all signatories, assign each a unique number within a 1-500 interval, then use a random number generator to pick 30 unique numbers. The individuals assigned those numbers will make up the random sample.

Step by step solution

01

Understand simple random sampling

Simple random sampling is a subset of a statistical population where each member of the subset has an equal probability of being chosen. A simple random sample is supposed to be an unbiased representation of a group.
02

Create a list

First, the president needs to prepare a comprehensive list of all 500 people who signed the petition.
03

Assign a unique number

Assign each individual on the list a unique number from 1 to 500.
04

Use a Random Number Generator

Then use a random number generator to generate 30 distinct numbers within this range. The generator should be set to not repeat numbers and only generate numbers within the 1-500 interval.
05

Select the Sample

Those individuals whose corresponding numbers were generated forms the simple random sample of 30 students whose signatures she checks with the registrar. This creates an equal likelihood of being selected for each of the petitioners, achieving a simple, random sample

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Key Concepts

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

Statistical Population
When it comes to understanding statistical population, it's like considering everyone at a big party. Imagine you're at the party and want to know how many people prefer chocolate cake over vanilla. The entire group of people at the party represents the statistical population. In the context of the student council president's task, the 500 signatures on the petition form the population - that is, all the individuals who could potentially be examined to answer her question about registered student status. Establishing an accurate population is crucial since it's the basis from which samples are drawn.
It's akin to knowing you have both chocolate and vanilla slices to serve before you can ask each guest their preference. Similarly, you need a complete list of all petitioners before you draw the sample.
Random Number Generator
Using a random number generator is like pulling raffle tickets from a hat. You want to give everyone an equal shot at winning, without any peeking! A random number generator is a tool that can pick numbers without any pattern or bias. In the scenario with our council president, after assigning each signature a unique number from 1 to 500, she'd use a random number generator to randomly select 30 unique numbers.
To ensure fairness, imagine if our raffle was rigged or the same numbers kept getting picked 鈥� it wouldn't be fair, right? That's why the random number generator must be set to not repeat numbers and stay within the limits (1 to 500), ensuring each signature has an equal chance of being chosen.
Sampling Methods
Imagine shopping for fruit and wanting a taste of the available mangoes. You randomly pick a few to try instead of tasting them all. This is a bit like sampling methods: the ways in which we can pick 'tastes' or samples from our bigger group - the population. Simple random sampling is one type, like selecting mangoes blindly from a basket.
There are others, too, such as stratified sampling, where you first divide the population into groups (like separating ripe and unripe mangoes) and then randomly sample within each group. Or cluster sampling, where you'd randomly pick whole boxes of mangoes rather than individual fruits. The choice of sampling method depends on the goals of your study and the nature of your population.
Unbiased Sample Representation
An unbiased sample representation is the golden ticket of sampling; it's what makes the results trustworthy. Think of it as a mirror reflecting the full variety of a crowd at a festival. If the mirror only shows people wearing blue hats, you'd get the wrong idea about what everyone else likes to wear.
An unbiased sample does its best to accurately reflect everyone in the population. It's like making sure that, in our earlier party scenario, both chocolate and vanilla cake fans are randomly asked about their preferences without leaning towards one group. The council president's sample of 30 signatures should ideally mirror the larger group of 500, giving a fair representation of the registered student status within the entire group.

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Most popular questions from this chapter

The article "Yes that Miley Cyrus Biography Helps Learning" (The Globe and Mail, August 5,2010 ) describes an experiment investigating whether providing summer reading books to low-income children would affect school performance. Subjects in the experiment were 1,300 children randomly selected from first and second graders at low-income schools in Florida. A group of 852 of these children were selected at random from the group of 1,300 participants to be in the book group. The other 478 children were assigned to the control group. Children in the book group were invited to a book fair in the spring to choose any 12 reading books that they could then take home. Children in the control group were not given any reading books, but were given some activity and puzzle books. These children received books each year for three years until the children reached third and fourth grade. The researchers then compared reading test scores of the two groups. a. Explain why randomly selecting 852 of the 1,300 children to be in the book group is equivalent to random assignment of the children to the two experimental groups. b. Explain the purpose of including a control group in this experiment.

1.32 In an experiment to compare two different surgical procedures for hernia repair ("A Single-Blinded, Randomized Comparison of Laparoscopic Versus Open Hernia Repair in Children," Pediatrics [2009]: 332-336), 89 children were assigned at random to one of the two surgical methods. The methods studied were laparoscopic repair and open repair. In laparoscopic repair, three small incisions are made, and the surgeon works through these incisions with the aid of a small camera that is inserted through one of the incisions. In the open repair, a larger incision is used to open the abdomen. One of the response variables was the amount of medication given after the surgery to control pain and nausea. The paper states, "For postoperative pain, rescue fentanyl \((1 \mathrm{mg} / \mathrm{kg})\) and for nausea, ondansetron \((0.1 \mathrm{mg} / \mathrm{kg})\) were given as judged necessary by the attending nurse blinded to the operative approach." a. Why do you think it was important that the nurse who administered the medications did not know which type of surgery was performed? b. Explain why it was not possible for this experiment to be double-blind.

In 2000 , the chairman of a California ballot initiative campaign to add "none of the above" to the list of ballot options in all candidate races was quite critical of a Field poll that showed his measure trailing by 10 percentage points. The poll was based on a random sample of 1,000 registered voters in California. He is quoted by the Associated Press (January 30,2000 ) as saying, "Field's sample in that poll equates to one out of 17,505 voters." This was so dishonest, he added, that Field should get out of the polling business! If you worked on the Field poll, how would you respond to this criticism?

Use the following information to answer. The paper "Turning to Learn: Screen Orientation and Reasoning from Small Devices" (Computers in Human Behavior [2011]: \(793-797\) ) describes a study that investigated whether cell phones with small screens are useful for gathering information. The researchers wondered if the ability to reason using information read on a small screen was affected by the screen orientation. The researchers assigned 33 undergraduate students who were enrolled in a psychology course at a large public university to one of two groups at random. One group read material that was displayed on a small screen in portrait orientation, and the other group read material on the same size screen but turned to display the information in landscape orientation (see the following figure). Is the study described an observational study or an experiment?

For each of the statistical studies described, indicate whether the study is an observational study or an experiment. Give a brief explanation for your choice. The paper "Health Halos and Fast-Food Consumption" (Journal of Consumer Research [2007]: 301-314) described a study in which 46 college students volunteered to participate. Half of the students were given a coupon for a McDonald's Big Mac sandwich and the other half were given a coupon for a Subway 12 -inch Italian BMT sandwich. (For comparison, the Big Mac has 600 calories, and the Subway 12 -inch Italian BMT sandwich has 900 calories.) The researchers were interested in how the perception of Subway as a healthy fast- food choice and McDonald's as an unhealthy fast-food choice would influence what additional items students would order with the sandwich. The researchers found that those who received the Subway coupon were less likely to order a diet soft drink, more likely to order a larger size drink, and more likely to order cookies than those who received the Big Mac coupon.
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