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Chapter 4: Problem 40

Suppose that \(35 \%\) of the registered voters in a state are registered as Republicans, \(40 \%\) as Democrats, and \(25 \%\) as Independents. A newspaper wants to select a sample of 1000 registered voters to predict the outcome of the next election. If they randomly select 350 Republicans, randomly select 400 Democrats, and randomly select 250 lndependents, did this sampling procedure result in a simple randon sanple of registered voters from this district? (a) Yes, because each registered voter had the same chance of being chosen. (b) Yes, because random chance was involved. (c) No, because not all registered voters had the same chance of being chosen. (d) No, because there were a different number of registered voters selected from each party. (e) No, because not all possible groups of 1000 registered voters had the same chance of being chosen.

Short Answer

Expert verified
(e) No, because not all possible groups of 1000 registered voters had the same chance of being chosen.

Step by step solution

01

Understand Simple Random Sampling

In a simple random sample, every member of the population has an equal chance of being selected. Additionally, every possible combination of members for the sample has an equal chance of being chosen.
02

Analyze the Given Sampling Procedure

The newspaper selects 350 Republicans, 400 Democrats, and 250 Independents. These numbers correspond directly to the percentage distribution of party registration, meaning voters are grouped by their party affiliation rather than being selected strictly at random.
03

Evaluate Equal Chances for Individuals

In this method, registered voters do not have an equal chance of being selected as individuals. Instead, selection is constrained to fixed counts for each party, so some individuals have no chance of selection if the quota for their affiliation is filled.
04

Evaluate Equal Chances for Combinations

This sampling method ensures that the specific numbers of each political group are chosen, so not all possible combinations of 1000 voters can be formed. This violates the principle of each combination of 1000 being equally likely in a simple random sample.
05

Conclude on Sampling Procedure

Since not all individuals and combinations have equal chances of selection, this is not a simple random sample.

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Key Concepts

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

Sampling Methods
Sampling methods are techniques used to select a portion of a population to make statistical inferences about the whole. There are several sampling methods, each with its distinct approach, such as:
  • Simple Random Sampling: Every member and every combination of members from the population has an equal chance of being selected.
  • Stratified Sampling: The population is divided into subgroups, or strata, and samples are drawn from each, maintaining representation across important categories.
  • Systematic Sampling: Members are selected at regular intervals from an ordered list.
  • Cluster Sampling: Entire groups, or clusters, are chosen randomly, rather than individuals.
These methods help ensure that the sample is representative of the population, making the findings of a study generalizable. Understanding different sampling methods is crucial for conducting effective and reliable research.
Statistical Sampling
Statistical sampling is a key part of statistical research, allowing researchers to gather data and draw conclusions about a whole population based on just a sample. The main idea is to collect a subset of data that reflects the larger group's characteristics, which helps in analyzing trends, testing hypotheses, and making predictions. Statistical sampling reduces the cost, time, and resources required compared to a full census.

There are various factors that make statistical sampling effective:
  • Sample Size: Larger samples tend to give more reliable results, but they must be manageable within resource constraints.
  • Sampling Frame: A complete list of the population from which the sample is drawn.
  • Sample Variability: Measures the dispersion or spread of the sample results, which can affect the confidence and precision of inferences made.
With properly planned statistical sampling, researchers can obtain accurate estimations of population parameters while facilitating efficient use of resources.
Probability Sampling
Probability sampling is a technique where every member of the population has a known, non-zero chance of being selected. It ensures fairness and randomness in the sampling process. Key types of probability sampling include:
  • Simple Random Sampling: As mentioned, each member has an equal chance of selection.
  • Stratified Sampling: Useful for ensuring representation of specific subgroups.
  • Cluster Sampling: Involves randomly selecting entire clusters instead of individual elements.
  • Systematic Sampling: Member selection occurs at regular intervals, starting from a randomly chosen point.
This method is beneficial because it minimizes selection bias and improves the likelihood that the sample accurately represents the population. The equal probability reduces bias, increases credibility, and often results in precise estimations when conducted properly.
Sample Selection
Sample selection is the process of deciding which members of a population will be included in a sample. Selecting a sample requires careful consideration to ensure it is representative of the broader group.
  1. Define the Population: Determine who you want to study, which provides clarity and direction.
  2. Choose a Sampling Method: Depending on the research goals, methods like random, stratified, or systematic sampling are considered.
  3. Determine Sample Size: Estimate how many subjects are needed to achieve reliable results.

Good sample selection results in data that more closely mirrors the population, leading to more accurate conclusions and better decision-making. It is essential to avoid common pitfalls such as sample bias, non-random selection processes, or poorly defined populations. Proper sample selection is the backbone of any effective study.

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Most popular questions from this chapter

In which of the circumstances below would you allow collecting personal information without the subjects' consent? (a) A government agency takes a random sample of income tax returns to obtain information on the average income of people in different occupations. Only the incomes and occupations are recorded from the returns, not the names. (b) A social psychologist attends public meetings of a religious group to study the behavior pattems of members. (c) A social psychologist pretends to be converted to membership in a religious group and attends private meetings to study the behavior patterns of members.

What proportion of students at your school use Twitter? To find out, you survey a simple random sample of students from the school roster. (a) Will your sample result be exactly the same as the true population proportion? Explain. (b) Which would be more likely to get your sample result closer to the true population value: an SRS of 50 students or an SRS of 100 students? Explain.

A group of students wants to perform an experiment to determine whether Brand \(A\) or Brand \(\mathrm{B}\) deodorant lasts longer. One group member suggests the following design: Recruit 40 student volunteers -20 male and 20 female. Separate by gender, because male and female bodies might respond differently to deodorant. Give all the males Brand A deodorant and all the females Brand B. Have each student rate how well the deodorant is still working at the end of the school day on a 0 to 10 scale. Then compare ratings for the two treatments. (a) Identify any flaws you see in the proposed design for this experiment. (b) Describe how you would design the experiment. Explain how your design addresses each of the problems you identified in part (a).

Will cash bonuses speed the return to work of unemployed people? A state department of labor notes that last year 68 % of people who filed claims for unemployment insurance found a new job within 15 weeks. As an experiment, this year the state offers 500 to people filing unemployment claims if they find a job within 15 weeks. The percent who do so increases to 77 %. What flaw in the design of this experiment makes it impossible to say whether the bonus really caused the increase? Explain.

A simple random sample of 1200 adult Americans is selected, and each person is asked the following question: "In light of the huge national deficit, should the government at this time spend additional money to establish a national system of health insurance?" Only \(39 \%\) of those responding answered "Yes." This survey (a) is reasonably accurate since it used a large simple random sample. (b) needs to be larger since only about 24 people were drawn from cach state. (c) probably understates the percent of people who favor a system of national health insurance. (d) is very inaccurate but neither understates nor overstates the percent of people who favor a system of national health insurance. Because simple random sampling was used, it is unbiased. (e) probably overstates the percent of people who favor a system of national health insurance.
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