Question

The study "Digital Footprints" (Pew Internet \& American Life Project, www.pewinternet.org, 2007) reported that \(47 \%\) of Internet users have searched for information about themselves online. The \(47 \%\) figure was based on a random sample of Internet users. Suppose that the sample size was \(n=300\) (the actual sample size was much larger). Answer the four key questions (QSTN) to confirm that the suggested method in this situation is a large-sample confidence interval for a population proportion.

Short answer

Determining whether to use a large-sample confidence interval for a population proportion depends on the quality of the data, the sample size being statistically significant, the clarity of the numbers/data, and the next steps based on the data. After considering these QSTN, we conclude that a large-sample confidence interval is an appropriate method to use here.

Step-by-step solution

Confirming Quality

Answer the question: Is the data reliable? Since the data comes from a reputable source, we can say that the quality of the data is trustworthy.

Check for Statistical Significance

Answer the question: Is the sample large enough? \nThe sample size is 300, which is a substantial size. Hence, we can say that the results attained could be statistically significant.

Considering Tangible Numbers

Answer the question: Are the numbers clear and detailed? The survey reported that 47% of Internet users have searched for information about themselves online. We have a tangible number, which is 47%.

Identify Next Steps

Answer the question : What should be done next based on the data? Based on the data, the next step would be to estimate the population proportion with a specific level of confidence, say for example 95% confidence level.

Determine method

Since the sample size is substantial, the data quality is good and there is a clear population proportion (47%), a large-sample confidence interval for a population proportion would be an appropriate method for further analysis.

Key concepts

Population Proportion

The notion of population proportion is fundamental when discussing survey results and statistical studies. It represents the percentage of the entire population that has a particular attribute or characteristic. In the context of the 'Digital Footprints' study, population proportion refers to the percentage of Internet users who have searched for information about themselves online.

For instance, the study cites that 47% of Internet users have done such a search. This figure is immensely significant because it does not pertain to only the individuals surveyed; it is an estimate for a much larger group - the global Internet userbase. Understanding this number is critical because it helps organizations and individuals make informed decisions based on the behaviors of a broader community.

However, to use this percentage for decision-making, one must ensure it is a reliable estimate of the true population proportion, which is where the concept of confidence intervals and adequate sample size comes into play.

Sample Size

Sample size, indicated by the variable 'n' in statistical formulas, is the number of observations or measurements taken from a population for analysis. The size of the sample is a crucial element because it directly influences the validity of the study's conclusions. A larger sample size generally leads to more reliable and accurate inferences about the population.

In the 'Digital Footprints' study, the sample size is mentioned as 300 Internet users. This number is considered large enough to use a large-sample confidence interval approach, a statistical method designed to estimate population parameters. The greater the sample size, the narrower and more precise the confidence interval becomes, which in turn implies a smaller margin of error and more faith in the representativeness of the sample.

Statistical Significance

Statistical significance is a term that's used to indicate whether the results of a study or experiment are not due to chance. It's a measure of how confident we can be in the results of our analysis. In studies like 'Digital Footprints,' demonstrating statistical significance ensures that the observed proportion genuinely reflects the behavior of the larger population, rather than being a random occurrence within the sample.

Typically, a result is deemed statistically significant if the p-value, a probability score, is below a predetermined threshold, such as 0.05 or 5%. If a study yields statistically significant results, policymakers, marketers, and other decision-makers can act on the findings with a degree of certainty that the effect or proportion observed truly exists in the broader population.

Data Reliability

Data reliability refers to the consistency and dependability of data collected during research. Reliable data must accurately represent what it's intended to measure, and the collection process must be free from biases and systematic errors.

In the solution of the 'Digital Footprints' study, reliability was confirmed because the data was sourced from a reputable organization known for its rigorous data collection standards. When data is reliable, like in this case, subsequent analyses, such as estimating a confidence interval, can be trusted to provide insights that are reflective of the actual population or phenomenon studied. The reliability of data is a cornerstone for credible research, guiding meaningful conclusions and informed decisions.