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Chapter 1: Problem 35

Is a sample of 500 volunteers a reliable measure for a population of 2,500?

Short Answer

Expert verified
Yes, a sample of 500 is reliable for a population of 2,500, provided it is randomly selected.

Step by step solution

01

Understanding the Concept of Sample Size

The reliability of a sample in representing a population depends on several factors such as the size of the sample, the total population size, and the variability within the population. We need to assess if the size of 500 is large enough to draw conclusions about a population of 2,500.
02

Calculating the Sample Proportion

Calculate the proportion that our sample represents of the total population. This is done by dividing the sample size by the total population size: \( \frac{500}{2500} = 0.2 \). This means the sample represents 20% of the population.
03

Evaluating Sample Size Adequacy

Common statistical standards suggest that a sample of at least 30 is required for a sample to approximate a normal distribution (central limit theorem). Additionally, a sample size above 10% of the population is typically considered good enough for reliable data, particularly if the sample is randomly selected and representative.
04

Conclusion on Reliability

Given that our sample of 500 represents 20% of the entire population of 2,500, which is well above the 10% threshold, and assuming it is randomly selected, it can be considered a reliable measure of the population.

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Key Concepts

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

Population Sampling
Population sampling is the method of selecting a subset of individuals from a larger group to examine certain characteristics or behaviors. This smaller group, called a sample, allows researchers to make inferences about the entire population.

One of the main reasons for using samples rather than studying an entire population is practicality. Studying an entire population can be costly, time-consuming, and sometimes impossible. By carefully selecting a sample, we can save resources while still obtaining valuable insights.
  • Random Sampling: Every member of the population has an equal chance of being selected. This reduces bias and ensures a representative sample.
  • Systematic Sampling: Selecting every nth member from a list of the population. While easier to implement, it's crucial that the list order doesn't influence the outcome.
  • Stratified Sampling: Dividing the population into subgroups, or strata, and randomly sampling from each. This ensures that each subgroup is represented in the sample.
Understanding these methods can help us determine if a chosen sample, like the 500 volunteers out of 2,500, is appropriately representing the entire group.
Central Limit Theorem
The central limit theorem (CLT) is a crucial concept in statistics. It states that, given a sufficiently large sample size, the sampling distribution of the sample mean will approximate a normal distribution, regardless of the population's original distribution.

This theorem is significant for several reasons:
  • It justifies the use of normal probability models in statistical testing, even if the data comes from a non-normal distribution.
  • It allows us to make probabilistic statements about sample means, which is particularly useful in confidence interval estimation and hypothesis testing.
The relevance of the CLT in determining our sample size of 500 lies in its assurance that this size is "sufficiently large" for most practical purposes, allowing us to confidently draw inferences about the population's characteristics.
Sample Proportion
Sample proportion refers to the fraction of the population that the sample represents. It is calculated by dividing the number of observations in the sample by the total number of observations in the population. In our example, the sample proportion is given by: \[ \frac{500}{2500} = 0.2 \] This means our sample represents 20% of the entire population.

Knowing the sample proportion is important because:
  • It helps us understand how closely the sample can reflect the population's features. A higher proportion generally yields a more accurate representation.
  • In combination with the sample size, it helps determine the margin of error, impacting the conclusions drawn from the sample data.
A 20% representation, as in this scenario, is quite substantial, especially when other conditions such as random selection are met, increasing the reliability of the findings.
Statistical Reliability
Statistical reliability refers to the consistency and dependability of a measure. A measure is considered reliable if it consistently produces similar results under consistent conditions. In the context of sampling, a sample is considered reliable when it accurately reflects the population's characteristics.

To achieve statistical reliability, consider the following:
  • Sample size: Larger samples tend to yield more reliable results, as they are likely to better capture the diversity within a population.
  • Randomness: Ensuring that the sample is randomly selected can enhance reliability by minimizing biases.
  • Proportion: As shown in the sample proportion of 20%, when the sample captures a significant portion of the population, it boosts the reliability of the statistical conclusions.
By ensuring our sample is representative and large enough, we can improve the statistical reliability of our study, making the insights drawn trustworthy and valid.

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Most popular questions from this chapter

Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience). The first 50 people who walk into a sporting event are polled on their television preferences.

A study was done to determine the age, number of times per week, and the duration (amount of time) of residents using a local park in San Jose. The first house in the neighborhood around the park was selected randomly and then every eighth house in the neighborhood around the park was interviewed. The sampling method was: a. simple random b. systematic c. stratified d. cluster

Use the following data to answer the next five exercises: A pair of studies was performed to measure the effectiveness of a new software program designed to help stroke patients regain their problem solving skills. Patients were asked to use the software program twice a day, once in the morning and once in the evening. The studies observed 200 stroke patients recovering over a period of several weeks. The first study collected the data in Table 1.31. The second study collected the data in Table 1.32. $$\begin{array}{|l|l|l|}\hline \text { Group } & {\text { Showed improvement }} & {\text { No improvement }} & {\text { Deterioration }} \\ \hline \text { Used program } & {142} & {43} & {15} \\ \hline \text { Did not use program } & {72} & {110} & {18} \\ \hline\end{array}$$ Table 1.31 $$\begin{array}{|l|l|l|}\hline \text { Group } & {\text { Showed improvement }} & {\text { No improvement }} & {\text { Deterioration }} \\ \hline \text { Used program } & {105} & {74} & {19} \\ \hline \text { Did not use program } & {89} & {99} & {12}\\\ \hline\end{array}$$ Table 1.32 Both groups that performed the study concluded that the software works. Is this accurate?

How does sleep deprivation affect your ability to drive? A recent study measured the effects on 19 professional drivers. Each driver participated in two experimental sessions: one after normal sleep and one after 27 hours of total sleep deprivation. The treatments were assigned in random order. In each session, performance was measured on a variety of tasks including a driving simulation. Use key terms from this module to describe the design of this experiment.

In advance of the 1936 Presidential Election, a magazine titled Literary Digest released the results of an opinion poll predicting that the republican candidate Alf Landon would win by a large margin. The magazine sent post cards to approximately 10,000,000 prospective voters. These prospective voters were selected from the subscription list of the magazine, from automobile registration lists, from phone lists, and from club membership lists. Approximately 2,300,000 people returned the postcards. a. Think about the state of the United States in 1936. Explain why a sample chosen from magazine subscription lists, automobile registration lists, phone books, and club membership lists was not representative of the population of the United States at that time. b. What effect does the low response rate have on the reliability of the sample? c. Are these problems examples of sampling error or non sampling error? d. During the same year, George Gallup conducted his own poll of 30,000 prospective voters. His researchers used a method they called "quota sampling" to obtain survey answers from specific subsets of the population. Quota sampling is an example of which sampling method described in this module?
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