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Chapter 1: Problem 15

Consider the population of all students at your school. A certain proportion support mandatory national service (MNS) following high school. Your friend randomly samples 20 students from the school, and uses the sample proportion who support MNS to predict the population proportion at the school. You take your own, separate random sample of 20 students, and find the sample proportion that supports MNS. a. For the two studies, are the populations the same? b. For the two studies, are the sample proportions necessarily the same? Explain.

Short Answer

Expert verified
a. Yes, the populations are the same. b. No, the sample proportions are not necessarily the same due to sampling variability.

Step by step solution

01

Identify the Populations

In both studies, the population refers to the entire group of students at your school. Hence, for the two studies, the populations are the same as they both aim to understand the same broader group's opinion on mandatory national service following high school.
02

Consider the Sampling Process

Each study involved taking a random sample of 20 students from the entire student population. Because both samples are drawn from the same overall population, these samples are subsets of the same group but composed of potentially different individuals.
03

Discuss the Sample Proportions

The sample proportion for each study is calculated based on whether the sampled students support MNS. While both samples are derived from the same population, the specific composition of each sample is different, meaning the sample proportions may differ due to random variability in the sampling process.
04

Conclude on Sample Proportions

The sample proportions for each of the two samples are not necessarily the same since both are influenced by the random selection of students. Variability in samples usually leads to differences in sample proportions even if they are derived from the same population.

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Key Concepts

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

Understanding Population and Sample
In any study, a population refers to the entire group of individuals that we're interested in learning about. In this case, the population is all the students at your school. When it's impractical to collect data from every individual in the population, we select a subset of that population, known as a sample. This sample should ideally be representative of the population. In our scenario, both you and your friend are interested in the opinion about mandatory national service (MNS) across the entire school, which means you are both working with the same population. It's important to recognize that a sample is only a part of the population, so conclusions drawn from a sample might not be perfectly representational of the broader group. Nevertheless, studying a sample rather than attempting to survey every student is often more practical and manageable.
Random Sampling Explained
Random sampling is a method used to obtain a sample in such a way that every individual in the population has an equal chance of being selected. This technique helps ensure that the sample is unbiased and representative of the population. Your study and your friend's study both used random samples of 20 students. By using random sampling, we aim to avoid biases that could sway the results of the survey in a particular direction. For example, if only students from a specific year or specific classes were chosen, it might not accurately represent the entire student body's views. Thanks to random sampling, each student, regardless of their opinions on MNS, has the same opportunity to be included in the sample.
The Concept of Sample Proportion
A sample proportion refers to the proportion of individuals in a sample that possess a particular characteristic. For instance, if 8 out of your 20 sampled students support MNS, the sample proportion would be 0.4 or 40%. Because you and your friend are studying the same population but taking separate samples, the sample proportion in each of your studies could differ. Even though both samples come from the same population, the random selection of students means that they might not have the exact same opinions. This variation is known as sampling variability and it exists in every random sampling process. Different samples can lead to different sample proportions simply due to chance. Hence, even with similar sample sizes, it's natural to see differing sample proportions in independent samples.

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