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Chapter 1: Problem 23

A university wants to determine what fraction of its undergraduate student body support a new \(\$ 25\) annual fee to improve the student union. For each proposed method below, indicate whether the method is reasonable or not. (a) Survey a simple random sample of 500 students. (b) Stratify students by their field of study, then sample \(10 \%\) of students from each stratum. (c) Cluster students by their ages (e.g. 18 years old in one cluster, 19 years old in one cluster, etc.), then randomly sample three clusters and survey all students in those clusters.

Short Answer

Expert verified
Methods (a) and (b) are reasonable; method (c) is not.

Step by step solution

01

Evaluate Simple Random Sampling

A simple random sample is generally a good approach as it gives each student an equal chance of being selected. In this case, surveying 500 students using this method is reasonable because it ensures a representative subset of the undergraduate student body.
02

Evaluate Stratified Sampling by Field of Study

Stratified sampling involves dividing the population into subgroups (strata) and sampling from each subgroup. Sampling 10% of students from each field of study ensures representation across different majors. This approach is reasonable as it accounts for possible differences in opinion based on academic field.
03

Evaluate Cluster Sampling by Age

Cluster sampling involves dividing the population into clusters, then selecting clusters randomly and surveying all members within them. Clustering by age may not accurately reflect opinions related to the fee since age isn't likely relevant to opinions about student union improvements. Surveying all students in three chosen clusters could inadvertently bias results, making this method unreasonable.

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Key Concepts

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

Simple Random Sampling
Simple random sampling is a fundamental statistical method used to gather data from a population. This technique ensures that every individual in the population has an equal chance of being selected. This randomness in selection is crucial because it helps to eliminate bias, leading to the collection of data that is representative of the broader group.

In our given example, the university would select 500 students at random out of all the undergraduates. By doing this, each student has the same opportunity to be part of the survey, which makes it fair and unbiased. The beauty of simple random sampling lies in its power to produce an unbiased sample if the sample size is large enough.

When conducting surveys like this, it鈥檚 important to have a sample that accurately represents the population. This method is suitable because it gives a snapshot of the entire student body鈥檚 opinion on the proposed annual fee for student union improvements.
Stratified Sampling
Stratified sampling is another compelling sampling method that can be especially useful when the population is diverse. This approach involves dividing the population into distinct subgroups, known as strata, based on specific characteristics or attributes. Then, a sample is taken from each stratum, often proportionate to its size within the population.

Consider the university's scenario: by stratifying students based on their field of study and selecting 10% from each group, we ensure that all study fields are represented. This is crucial because students from different disciplines might have varied perspectives on the proposed fee increase. By using stratified sampling, the university can obtain a wide range of opinions and ensure that no particular group dominates the results. This method can yield more precise estimates than simple random sampling when there are known variations between strata.
Cluster Sampling
Cluster sampling is particularly useful in situations where the population is spread out geographically or when a list of the entire population is not available. This method involves dividing the population into clusters, which are ideally mini-representations of the entire population. After that, a few clusters are randomly selected, and all individuals within those clusters are surveyed.

However, the effectiveness of cluster sampling depends greatly on how these clusters are formed. In the exercise at hand, clustering by age might not be very effective since a student's age may not correlate with their opinion about the student union fee. This could lead to skewed results if the clusters don't adequately represent the entire student body's views.

While cluster sampling can be cost-effective and logistically feasible, it requires careful consideration of the factors used to form clusters to avoid bias and ensure that survey results are meaningful.

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Most popular questions from this chapter

A study that surveyed a random sample of otherwise healthy high school students found that they are more likely to get muscle cramps when they are stressed. The study also noted that students drink more coffee and sleep less when they are stressed. (a) What type of study is this? (b) Can this study be used to conclude a causal relationship between increased stress and muscle cramps? (c) State possible confounding variables that might explain the observed relationship between increased stress and muscle cramps.

The Stanford Open Policing project gathers, analyzes, and releases records from traffic stops by law enforcement agencies across the United States. Their goal is to help researchers, journalists, and policymakers investigate and improve interactions between police and the public. \(^{47}\) The following is an excerpt from a summary table created based off of the data collected as part of this project. $$ \begin{array}{lllrrr} \hline & & \text { Driver's } & \text { No. of stops } & {\text { \% of stopped }} \\ \text { County } & \text { State } & \text { race } & \text { per year } & \text { cars searched } & \text { drivers arrested } \\ \hline \text { Apaice County } & \text { Arizona } & \text { Black } & 266 & 0.08 & 0.02 \\ \text { Apaice County } & \text { Arizona } & \text { Hispanic } & 1008 & 0.05 & 0.02 \\ \text { Apaice County } & \text { Arizona } & \text { White } & 6322 & 0.02 & 0.01 \\ \text { Cochisc County } & \Lambda \text { rizona } & \text { Black } & 1169 & 0.05 & 0.01 \\ \text { Cochise County } & \text { Arizona } & \text { Hispanic } & 9453 & 0.04 & 0.01 \\ \text { Cochise County } & \text { Arizona } & \text { White } & 10826 & 0.02 & 0.01 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ \text { Wood County } & \text { Wisconsin } & \text { Black } & 16 & 0.24 & 0.10 \\ \text { Wood County } & \text { Wisconsin } & \text { Hispanic } & 27 & 0.04 & 0.03 \\ \text { Wood County } & \text { Wisconsin } & \text { White } & 1157 & 0.03 & 0.03 \\ \hline \end{array} $$ (a) What variables were collected on each individual traffic stop in order to create to the summary table above? (b) State whether each variable is numerical or categorical. If numerical, state whether it is continuous or discrete. If categorical, state whether it is ordinal or not. (c) Suppose we wanted to evaluate whether vehicle search rates are different for drivers of different races. In this analysis, which variable would be the response variable and which variable would be the explanatory variable?

The Buteyko method is a shallow breathing technique developed by Konstantin Buteyko, a Russian doctor, in 1952. Anecdotal evidence suggests that the Buteyko method can reduce asthma symptoms and improve quality of life. In a scientific study to determine the effectiveness of this method, researchers recruited 600 asthma patients aged \(18-69\) who relied on medication for asthma treatment. These patients were randomnly split into two research groups: one practiced the Buteyko method and the other did not. Patients were scored on quality of life, activity, asthma symptoms, and medication reduction on a scale from 0 to 10 . On average, the participants in the Buteyko group experienced a significant reduction in asthma symptoms and an improvement in quality of life. \({ }^{13}\) (a) Identify the main research question of the study. (b) Who are the subjects in this study, and how many are included? (c) What are the variables in the study? Identify each variable as numerical or categorical. If numerical, state whether the variable is discrete or continuous. If categorical, state whether the variable is ordinal.

A city council has requested a household survey be conducted in a suburban area of their city. The area is broken into many distinct and unique neighborhoods, some including large homes, some with only apartments, and others a diverse mixture of housing structures. For each part below, identify the sampling methods described, and describe the statistical pros and cons of the method in the city's context. (a) Randomly sample 200 households from the city. (b) Divide the city into 20 neighborhoods, and sample 10 households from each neighborhood. (c) Divide the city into 20 neighborhoods, randomly sample 3 neighborhoods, and then sample all households from those 3 neighborhoods. (d) Divide the city into 20 neighborhoods, randomly sample 8 neighborhoods, and then randomly sample 50 households from those neighborhoods. (e) Sample the 200 households closest to the city council offices.

Suppose we want to estimate household size, where a "household" is defined as people living together in the same dwelling, and sharing living accommodations. If we select students at random at an elementary school and ask them what their family size is, will this be a good measure of household size? Or will our average be biased? If so, will it overestimate or underestimate the true value?
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