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Chapter 10: Problem 5

Zane is interested in the proportion of people who recycle each of three distinct products: paper, plastic, electronics. He wants to test the hypothesis that the proportion of people recycling each type of product differs by age group: \(12-18\) years old, \(19-30\) years old, \(31-40\) years old, over 40 years old. Describe the sampling method appropriate for a test of homogeneity regarding recycled products and age.

Short Answer

Expert verified
Use stratified random sampling by age group to ensure each age category is represented in the recycling behavior study.

Step by step solution

01

Understanding the Test of Homogeneity

A test of homogeneity is used to determine if different populations have the same distribution of a categorical variable. In this context, we want to compare the recycling habits across different age groups for three types of products: paper, plastic, and electronics.
02

Define the Population and Samples

The populations are the different age groups (12-18, 19-30, 31-40, and over 40 years old). For each age group, a sample needs to be collected to represent the recycling behavior of that age category. Each sample should include individuals from that particular age group who recycle paper, plastic, or electronics.
03

Choose a Random Sampling Method

To ensure that the samples from each age group are representative of that group, a random sampling method should be utilized. This could be a simple random sample where individuals are randomly chosen from each age category, or a stratified random sample where the population is divided into strata (age groups) and random samples are taken from each stratum.
04

Ensure Adequate Sample Sizes

For a valid test of homogeneity, all samples should have adequate sizes to ensure meaningful comparison. Each age group (stratum) should have similar sample sizes or proportional sizes to their population distribution to ensure robust results.

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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 essential techniques in statistics that help in selecting a group of subjects from a larger population, enabling researchers to gather data.
In the context of Zane's study on recycling habits, an appropriate sampling method is crucial for a valid test of homogeneity, which aims to determine if distribution differences exist among age groups. Sampling ensures that conclusions drawn from the sample can be generalized to the larger population with a degree of certainty.
  • **Simple Random Sampling**: Every individual has an equal chance of being selected. This method is straightforward and minimizes selection bias.
  • **Stratified Sampling**: The population is divided into homogeneous sub-groups, or strata, and samples are drawn from each stratum. This method ensures that specific segments of the population are adequately represented, which is critical when analyzing differences among age groups.
Each of these methods is chosen based on the research objective and the nature of the population being studied.
Random Sampling
Random sampling is a fundamental element in ensuring that the samples collected are representative of the entire population. This method is pivotal in reducing bias, thus leading to more accurate and reliable results in studies like Zane's on recycling habits across age groups.
Ensuring randomness means every member of the population has an equal chance of being included in the sample. This is often achieved through methods like drawing names from a hat or using random number generators. The randomness helps in:
  • Providing a clear picture of population behavior, avoiding patterns or biases associated with non-random methods.
  • Facilitating the generalization of results to the wider population, as the findings reflect true population characteristics more closely.
Random sampling can be simple or involve stratifying before random selection, depending on study requirements.
Stratified Sampling
Stratified sampling is a technique where the population is divided into distinct subgroups known as strata. In Zane's recycling study, age groups act as these strata. It ensures that key demographic groups are proportionally represented in the sample.
Here's how it works:
  • The population is categorized into mutually exclusive segments that are relevant to the study 鈥� in this case, age groups.
  • A random sample is then drawn from each stratum, ensuring that each subgroup is represented in the final sample.
  • This method controls for variability within the strata, leading to more reliable comparison across groups.
By using stratified sampling, researchers like Zane can account for potential variations in recycling habits between different age brackets, thereby conducting a more thorough and accurate analysis.
Population Distribution
Population distribution refers to how different segments of a population are spread out. In the context of Zane's study, it involves understanding how recycling practices differ among the distinct age groups involved in the research.
Understanding population distribution is key because:
  • It helps to identify the number of individuals within each age group for forming strata, ensuring that sample sizes reflect the true proportions of these groups in the population.
  • Gives insights into demographic patterns which might influence recycling behaviors, like varying awareness levels or access to recycling facilities across ages.
Accurate knowledge of population distribution also allows researchers to make sure that the samples drawn are reflective of the real-world demographics they aim to study, minimizing errors in hypothesis testing.
Age Groups
Different age groups often display varied behaviors and attitudes towards practices like recycling. In Zane's study, these categories are essential for understanding the differences in recycling habits.
Different age brackets (12-18, 19-30, 31-40, and over 40) allow insights into:
  • The influence of generational changes on environmental practices.
  • How awareness and accessibility to recycling can shift across ages.
  • The social and economic factors that might affect recycling habits in each age group.
Knowing age-based trends enables targeted initiatives and policies to boost recycling rates by addressing the specific needs and obstacles encountered by each group.

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Most popular questions from this chapter

A new thermostat has been engineered for the frozen food cases in large supermarkets. Both the old and the new thermostats hold temperatures at an average of \(25^{\circ} \mathrm{F}\). However, it is hoped that the new thermostat might be more dependable in the sense that it will hold temperatures closer to \(25^{\circ} \mathrm{F}\). One frozen food case was equipped with the new thermostat, and a random sample of 21 temperature readings gave a sample variance of \(5.1 .\) Another, similar frozen food case was equipped with the old thermostat, and a random sample of 16 temperature readings gave a sample variance of \(12.8 .\) Test the claim that the population variance of the old thermostat temperature readings is larger than that for the new thermostat. Use a \(5 \%\) level of significance. How could your test conclusion relate to the question regarding the dependability of the temperature readings?

Explain why goodness-of-fit tests are always right-tailed tests.

The following table shows ceremonial ranking and type of pottery sherd for a random sample of 434 sherds at a location in the Sand Canyon Archaeological Project, Colorado (The Architecture of Social Integration in Prehistoric Pueblos, edited by Lipe and Hegmon). $$\begin{array}{lcccc} \hline \begin{array}{l} \text { Ceremonial } \\ \text { Ranking } \end{array} & \begin{array}{c} \text { Cooking Jar } \\ \text { Sherds } \end{array} & \begin{array}{c} \text { Decorated Jar Sherds } \\ \text { (Noncooking) } \end{array} & \text { Row Total } \\ \hline \mathrm{A} & 86 & 49 & 135 \\ \hline \mathrm{B} & 92 & 53 & 145 \\ \hline \mathrm{C} & 79 & 75 & 154 \\ \hline \text { Column Total } & 257 & 177 & 434 \\ \hline \end{array}$$ Use a chi-square test to determine if ceremonial ranking and pottery type are independent at the 0.05 level of significance.

Rothamsted Experimental Station (England) has studied wheat production since \(1852 .\) Each year, many small plots of equal size but different soil/fertilizer conditions are planted with wheat. At the end of the growing season, the yield (in pounds) of the wheat on the plot is measured. The following data are based on information taken from an article by G. A. Wiebe in the Journal of Agricultural Research (Vol. \(50,\) pp. \(331-357\) ). For a random sample of years, one plot gave the following annual wheat production (in pounds): $$\begin{array}{lllllll} 4.15 & 4.21 & 4.27 & 3.55 & 3.50 & 3.79 & 4.09 & 4.42 \\ 3.89 & 3.87 & 4.12 & 3.09 & 4.86 & 2.90 & 5.01 & 3.39 \end{array}$$ Use a calculator to verify that, for this plot, the sample variance is \(s^{2} \approx 0.332\) Another random sample of years for a second plot gave the following annual wheat production (in pounds): $$\begin{array}{llllllll} 4.03 & 3.77 & 3.49 & 3.76 & 3.61 & 3.72 & 4.13 & 4.01 \\ 3.59 & 4.29 & 3.78 & 3.19 & 3.84 & 3.91 & 3.66 & 4.35 \end{array}$$ Use a calculator to verify that the sample variance for this plot is \(s^{2} \approx 0.089\) Test the claim that the population variance of annual wheat production for the first plot is larger than that for the second plot. Use a \(1 \%\) level of significance.

Does the \(x\) distribution need to be normal in order to use the chi-square distribution to test the variance? Is it acceptable to use the chi-square distribution to test the variance if the \(x\) distribution is simply mound- shaped and more or less symmetric?
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