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Authors of the news release titled "Major Gaps Still Exist Between the Perception and the Reality of Americans' Internet Security Protections, Study Finds" (The National Cyber Security Alliance) estimated the proportion of Americans who claim to have a firewall installed on their computer to protect them from computer hackers to be .80 based on a survey conducted by the Zogby market research firm. They also estimated the proportion of those who actually have a firewall installed to be .42, based on checkups performed by Norton's \(\mathrm{PC}\) Help software. The following quote is from the news release: For the study, NCSA commissioned a Zogby survey of more than 3000 Americans and Symantec conducted checkups of 400 Americans' personal computers performed by \(\mathrm{PC}\) Help by Norton (www.norton.com/tuneup). The Zogby poll has a margin of error of \(+1-1.6 \%\) and the checkup has a margin of error of \(+1-5 \%\). Explain why the margins of error for the two estimated proportions are different.

Step by step solution

01

Understanding Margins of Error

The margin of error in a study or survey is an estimate of how much results are likely to differ from the true population value. It is fundamentally a reflection of the precision of an estimate, which is influenced by the study design, particularly the sample size.

02

Determine Sample Sizes

In this study, the Zogby poll, which had an estimated proportion of 0.8, surveyed 3000 Americans, while the Norton's PC help software, which had an estimated proportion of 0.42, only checked 400 American personal computers.

03

Analyze the Impact of Sample Size

Generally, as the sample size increases, the margin of error decreases. This is because collecting more data points provides more information about the population and hence makes the estimate more precise. Consequently, the Zogby poll, which had a larger sample size, had a smaller margin of error (±1.6%) compared to the Norton's PC Help software checkup, which had a smaller sample size and correspondingly a larger margin of error (±5%).

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Key Concepts

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

Survey Sampling

Survey sampling is the process of selecting a subset of individuals from a target population to estimate characteristics of the whole group. Since it's usually impractical or impossible to study the entire population, researchers draw a sample that is meant to represent the population as accurately as possible.

In the case of the Zogby survey, out of the vast number of Americans using the internet, 3000 individuals were selected to participate in the study to estimate the proportion of those with firewall protection. The key to high-quality survey sampling is to ensure the sample is random and representative, meaning every member of the population has an equal chance of being included, and the sample reflects the diversity of the population well. These principles help in reducing sampling bias and improving the reliability of the survey results.

Sample Size

Sample size, simply put, is the number of participants or observations included in a study. The size of the sample plays a critical role in the accuracy and reliability of the survey results. The larger the sample, the closer the results are likely to reflect the true characteristics of the population. However, bigger samples also mean more resources in terms of time, money, and effort.

In our example, the Zogby survey's sample size of 3000 participants likely contributes to its smaller margin of error when compared to Norton's PC Help checkup, which had a sample size of 400. Larger sample sizes typically result in a smaller margin of error because they offer a clearer picture of the population's traits, minimizing the uncertainty of the estimates.

Statistical Precision

Statistical precision refers to the closeness of repeated estimates to the true value they are supposed to measure. It represents the consistency and reliability of the result, based on the variation or spread of data points gathered in the survey. A higher precision denotes less variability and suggests that repeated measurements would yield similar results.

When a survey has a smaller margin of error, like in the Zogby poll with ±1.6%, it indicates that the results are more precise and there is higher confidence that the survey's estimate of 0.8 for the proportion of Americans with firewalls is close to the true proportion in the overall population. The larger margin of error in the Norton's PC Help software checkup (±5%) suggests that the estimate of 0.42 is less precise and there is more uncertainty about how closely this figure represents the true stats of the population.

Population Estimation

Population estimation is a statistical method used to infer the characteristics of a larger group based on sampled data. It involves using various techniques to predict broader trends or totals from the sample data. The accuracy of population estimation is heavily dependent on both the quality of the sample and the method used for estimation. If the sample is not representative of the population, the estimation could be biased and misleading.

In the internet security study, the researchers estimated the proportion of Americans with firewalls based on their respective samples. They did this using surveys and technological checkups. The results, however, should be interpreted with caution due to the different margins of error, which are reflective of the varying degrees of certainty around the population estimates derived from the two different sample sizes and methodologies.